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+{"text":"\\section*{Appendix A:  Three-qubit bit-flip code}\nIn this section, we explain how the deviation of the physical GKP qubits is projected onto the deviation of the ancillae. Fig.\\ref{sfiga} shows a quantum circuit for the QEC with the three-qubit bit-flip code. This circuit looks almost the same as the circuit for DV apart from the third ancilla qubit. However, the error identification for the GKP qubits is substantially different from that for DV-QEC. In this circuit, the sum of deviations of the physical GKP qubits $i$ and $i+1$ ($i = 1, 2$) are projected onto the ancilla $i$. The deviation of the physical GKP qubit 3 is projected onto ancilla 3. First, a single logical qubit $\\ket{\\widetilde{\\psi}}_{{\\rm L}}$ is prepared by two controlled-not (CNOT) gates acting on the data qubit $\\ket{\\widetilde{\\psi}}_{\\rm 1}$ = $\\alpha \\ket{\\widetilde{0}}_{\\rm 1}$ +$\\beta \\ket{\\widetilde{1}}_{\\rm 1}$ and two ancillae $\\ket{\\widetilde{0}}_{i}$ ( $i$ = 2, 3). The CNOT gate, which corresponds to the operator exp(-$i\\hat{q}_{\\rm L}\\hat{p}_{\\rm A}$), transforms\n\\begin{align*}\n\\hat{q}_{\\rm L} \\to   \\hat{q}_{\\rm L}, \\ \\ \n\\hat{p}_{\\rm L} \\to  \\hat{p}_{\\rm L} - \\hat{p}_{\\rm A} ,  \\nonumber \\\\\n\\hat{q}_{\\rm A} \\to   \\hat{q}_{\\rm A} +  \\hat{q}_{\\rm L}, \\ \\\n\\hat{p}_{\\rm A} \\to  \\hat{p}_{\\rm A},\n\\tag{A1}\n\\end{align*}\nwhere $\\hat{q_{\\rm L}}$ ($\\hat{q_{\\rm A}}$) and $\\hat{p_{\\rm L}}$ ($\\hat{p_{\\rm A}}$) are the $q$ and $p$ quadrature operators of the logical (ancilla) qubit, respectively. Then, the GQC displaces the $q$ and $p$ quadratures randomly and independently, and increases the variance of the three physical GKP qubits. After the GQC, the bit-flip error correction is implemented using the three ancillae $\\ket{\\widetilde{0}}_{{\\rm A}j}$ ($j$=1, 2) and $\\ket{\\widetilde{+}}_{{\\rm A}3}$. Before the CNOT gates in the error correction circuit, the true deviation values of the physical GKP qubits and ancillae in $q$ quadrature, which obey Gaussian distribution with mean zero, are denoted by $\\overline{\\Delta}_{j}$ and $\\overline{\\Delta}_{{\\rm A}j}$ ($j$=1, 2, 3), respectively. For simplicity, because the ancilla qubits are fresh, we assume that the initial variance is much smaller than that of the physical qubits of the logical qubit. Then, the CNOT gates change the true deviation values of three ancillae $\\overline{\\Delta}_{{\\rm A}j}$ in $q$ quadrature as follows:\n\\begin{figure}[t]\n    \\includegraphics[angle=270, width=1.0\\columnwidth]{sfiga.ps}\n    \\caption{\\label{sfiga}A quantum circuit of the QEC for the three-qubit bit-flip code with GKP qubits using the proposed method. The data qubit $\\ket{\\widetilde{\\psi}}_{\\rm 1}$ and two GKP qubits $\\ket{\\widetilde{0}}_{\\rm 2}$ and $\\ket{\\widetilde{0}}_{\\rm 3}$ encode a single logical qubit. $\\ket{\\widetilde{0}}_{\\rm A1}, \\ket{\\widetilde{0}}_{\\rm A2}, {\\rm and} \\ket{\\widetilde{+}}_{\\rm A3}$ denote ancilla qubits for the QEC. The GQC and $M_{q}$ denote the GQC and measurements of ancillae in $q$ quadrature, respectively.}   \n\\end{figure}\n\\begin{align*}\n\\overline{\\Delta}_{{\\rm A1}}  \\to   \\overline{\\Delta}_{{\\rm A1}} + \\overline{\\Delta}_{{\\rm 1}} + \\overline{\\Delta}_{{\\rm 2}}= \\overline{\\Delta}_{{\\rm 1}} + \\overline{\\Delta}_{{\\rm 2}}\\ \\nonumber ,\\\\\n \\overline{\\Delta}_{{\\rm A2}}  \\to   \\overline{\\Delta}_{{\\rm A2}} + \\overline{\\Delta}_{{\\rm 2}} + \\overline{\\Delta}_{{\\rm 3}}=\\overline{\\Delta}_{{\\rm 2}} + \\overline{\\Delta}_{{\\rm 3}}\\ \\nonumber , \\\\\n \\overline{\\Delta}_{{\\rm A3}} \\to  \\overline{\\Delta}_{{\\rm A3}} + \\overline{\\Delta}_{{\\rm 3}} = \\overline{\\Delta}_{{\\rm 3}} .\n \\tag{A2}\n\\end{align*}\nTherefore, the sum of deviations of the physical GKP qubits $i$ and $i+1$ ($i = 1, 2)$ are projected onto the ancilla $i$. The deviation of physical GKP qubit 3 is projected onto ancilla 3.\n\n\\section*{Appendix B: $C_{4}\/C_{6}$ code}\nThe error correction in the $C_{4}\/C_{6}$ code is based on quantum teleportation, where the logical qubit $\\ket{\\widetilde{\\psi}}_{{\\rm L}}$ encoded by the $C_{4}\/C_{6}$ code is teleported to the fresh encoded Bell state, as shown in Fig.\\ref{sfigb}. The quantum teleportation process refers to the outcomes $M_{p}$ and $M_{q}$ of the Bell measurement on the encoded qubits, and determines the amount of displacement. We obtain the Bell measurement outcomes of bit values $m_{pi}$ and $m_{qi}$ for the $i$-th physical GKP qubit of the encoded data qubit and encoded qubit of the encoded Bell state, respectively. In addition to bit values, we also obtain deviation values $\\Delta_{p{\\rm m}i}$ and $\\Delta_{q{\\rm m}i}$ for the $i$-th physical GKP qubit. Therefore, the proposed likelihood method can improve the error tolerance of the Bell measurement.\n\nAs a simple example to explain our method for the Bell measurement, we describe the level-1 $C_{4}\/C_{6}$ code, that is, the $C_{4}$ code. The $C_{4}$ code is the $[[\\rm{4,2,2}]]$ code and consists of four physical GKP qubits to encode a level-1 qubit pair; thus, it is not the error-correcting code but the error-detecting code in the conventional method. The logical bit value of the $C_{4}$ code is $k$ (=0,1) when the bit value of the level-1 qubit pair is ($k$,0) or ($k$,1), that is, the bit value of the first qubit $k$ defines a logical bit value of a qubit pair. As the parity check of the $Z$ operator for the first and second qubits $ZIZI$ and $IIZZ$ indicates, the bit value of the level-1 qubit pair (0,0) corresponds to the bit value of the physical GKP qubits $(m_{q{\\rm1}}, m_{q{\\rm 2}}, m_{q{\\rm 3}}, m_{q{\\rm 4}})$ = (0,0,0,0) or (1,1,1,1)~\\cite{Knill}. The bit values of the pairs (0,1), (1,0), and (1,1) correspond to the bit values of the physical GKP qubits (0,1,0,1) or (1,0,1,0), (0,0,1,1) or (1,1,0,0), and (0,1,1,0) or (1,0,0,1), respectively. Therefore, if the measurement outcome of the physical GKP qubits is (0,0,1,0) for the $Z$ basis, then we consider two error patterns, assuming the level-1 qubit pair (0,0). The first pattern is a single error on the physical qubit 3 and the second pattern is the triple errors on the physical qubits 1, 2, and 4. We then calculate the likelihood for the level-1 qubit pair (0,0) $F_{0,0}$ as\n\\begin{widetext}\n\\begin{equation*}\nF_{0 ,0} = f(\\sqrt{\\pi}-|\\Delta_{q{\\rm m1}}|)f(\\sqrt{\\pi}-|\\Delta_{q{\\rm m2}}|)f(\\Delta_{q{\\rm m3}})f(\\sqrt{\\pi}-|\\Delta_{q{\\rm m4}}|)\n                 + f(\\Delta_{q{\\rm m1}})f(\\Delta_{q{\\rm m2}})f(\\sqrt{\\pi}-|\\Delta_{q{\\rm m3}}|)f(\\Delta_{q{\\rm m4}}).\\tag{B1}\n\\end{equation*}\n\\end{widetext} \nWe similarly calculate the $ F_{0,1}, F_{1,0},$ and $F_{1,1}$ likelihood for the bit value of qubit pairs (0,1), (1,0), and (1,1). Finally, we determine the level-1 logical bit value for the $Z$ basis by comparing $F_{0,0}+F_{0,1}$ with $F_{1,0}+F_{1,1}$, which refer to the likelihood functions for the logical bit values zero and one, respectively. If $F_{0,0}+F_{0,1} > F_{1,0}+F_{1,1}$, then we determine that the level-1 logical bit value for the $Z$ basis is zero, and vice versa. The level-1 logical bit value for the $X$ basis can be determined by the parity check of the $X$ operator for the first and second qubits $XXII$ and $IXIX$ in a similar manner. In the conventional likelihood method~\\cite{Pou,Goto} $F_{0,0}$, $F_{0,1}$, $F_{1,0}$, and $F_{1,1}$ are given by the same joint probability\n\\begin{equation*}\np_{\\rm corr}^{3} (1-p_{\\rm corr}) + p_{\\rm corr}(1-p_{\\rm corr})^{3},\\tag{B2}\n\\end{equation*}\n\\begin{figure}[b]\n    \\includegraphics[angle=270, width=1.0\\columnwidth]{sfigb.ps}  \n    \\caption{\\label{sfigb}Error correction by quantum teleportation. The encoded data qubit $\\ket{\\widetilde{\\psi}}_{\\rm L}$, two encoded qubits $\\ket{\\widetilde{+}}_{\\rm L}$, and $\\ket{\\widetilde{0}}_{\\rm L}$ are encoded by $C_{4}\/C_{6}$ code. GQC and  MLD denote the GQC and a maximum-likelihood decision, respectively.}\n\\end{figure}\nwhere the probability $p_{corr}$ is defined by Eq. (1) in the main text.\nBecause $F_{0,0}+F_{0,1} = F_{1,0}+F_{1,1}$, the $C_{4}$ code is not error-correcting code but error-detecting code in the conventional method, whereas it is the error-correcting code in our method. For higher levels of concatenation, the likelihood for the level-$l$ ($l\\geqq2$) bit value can be calculated by the likelihood for the level-$(l-1)$ bit value in a similar manner.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Plane curves approximating varieties near the tangent cone.}\\label{s:pc}\n\n In this section, we generalize the notion of multiplicity of intersection\n used in \\cite[Chapter~3,~\\S4, Definition~1]{clo}.  Proposition~3 in\n \\cite[Chapter~9,~\\S6]{clo},\n has a criterion which says that a line lies in the tangent space\n of the variety iff the line meets the variety with multiplicity at least 2.\n Here, we show that through a singular point of an affine variety\n we can always draw a smooth plane curve inside the ambient affine space,\n which has the multiplicity of intersection with the variety at least 3.\n This basic result does not seem to appear anywhere in\n the literature.\n\n\n Let $K$ be an arbitrary field.\n An algebraic curve $C\\subset{\\Bbb A}^{n}$ can be\n parameterized at its smooth point $p$ by Taylor's series\n $$G(t)=p+t\\cdot v_1+t^2\\cdot v_2+\\cdots,$$\n where $t$ is a local parameter.\n Using this parameterization, we can define multiplicity of intersection\n of an affine variety with the curve $C$ at the point $p$.\n\n \\begin{defn}\n Let $m$ be a nonnegative integer and $X\\subset{\\Bbb A}^{n}$\n an affine variety, given by\n $\\text{\\bf I}(X)\\subset K[x_1,\\ldots,x_n]$. Suppose that we have\n a curve $C$, smooth at a point $p$ and with the parameterization as above.\n Then $C$ {\\bf meets} $X$ {\\bf with multiplicity} $m$  at $p$\n if $t=0$ is a zero of multiplicity at least $m$ of the series\n $f\\circ G(t)$  for all $f\\in\\text{\\bf I}(X)$, and for some $f$, of multiplicity\n exactly $m$. We denote this multiplicity of intersection by $I(p,C,X)$.\n \\end{defn}\n\n \\begin{rem}\n This definition does not depend on the choice of the parameterization.\n To check this one can use the following criterion:\n $t=0$ is a zero of multiplicity $m$ of $h(t)\\in K[[t]]$ if and only if\n $h(0)=h'(0)=\\cdots=h^{(m-1)}(0)=0$ but $h^{(m)}(0)\\ne 0$.\n \\end{rem}\n\n\n\n\n Also, we denote by $T_{p}X$ ($TC_{p}X$)\n the tangent space (cone) of a variety $X$ at a point $p$.\n The next result contains a necessary condition for a line to be in the tangent\n cone.\n Unfortunately, it is not a sufficient one.\n\n \\begin{thm}\\label{t:m}\n Let $X\\subset{\\Bbb A}^{n}$ be an affine variety and $p\\in X$ a singular\n point. Then for each $v\\in TC_{p}X$ there exists a smooth\n curve $C\\subset{\\Bbb A}^{n}$ through $p$ such that\n $v\\in T_{p}C$ and $I(p,C,X)\\geq3$.\n \\end{thm}\n\n \\begin{pf} Let  $v\\in TC_{p}X$.\n We can choose affine coordinates $x_{1},\\ldots,x_{n}$, so that\n $p=(0,\\ldots,0)$. Let $C$ be a curve through\n $p$ parameterized\n by $G(t)=t\\cdot v+t^{2}\\cdot\\gamma$, where\n $v\\in TC_{p}X$,\n $\\gamma=(\\gamma_1,\\ldots,\\gamma_{n})$.\n It suffices  to show that there is $\\gamma$, which is  zero or not a scalar\n multiple\n of $v$, such that $t=0$ is a zero\n of multiplicity $\\geq3$ of $f\\circ G(t)$  for all $f\\in\\text{\\bf I}(X)$.\n In this case, $C$ will be a smooth curve with the local parameter $t$\n at $p$.\n\n For each $f\\in\\text{\\bf I}(X)$, denote by\n $l_{f}(x)=l_{1}x_{1}+\\cdots+l_{n}x_{n}$\n  and\n $q_{f}(x)$ the  linear and quadratic parts of $f$.\n According to this notation we define\n $$W=\\{(l_{1},\\ldots,l_{n},q_{f}(v)):\\,f\\in\\text{\\bf I}(X)\\}\\subset\n k^{n+1},$$ which is a  subspace  since $\\text{\\bf I}(X)$ is. For all\n $f\\in\\text{\\bf I}(X)$,    we have\n $$f(t\\cdot v+t^{2}\\cdot\\gamma)=l_{f}(t\\cdot v)+l_{f}(t^{2}\\cdot\\gamma)+\n q_{f}(t\\cdot v)+\\text{ terms of degree }\\geq3\\text{ in }t.$$\n Since $v\\in T_{p}X$, we get $l_{f}(v)=0$ for any $f\\in\\text{\\bf I}(X)$.\n Hence,\n $$f(t\\cdot v+t^{2}\\cdot\\gamma)\\equiv\n t^{2}(l_{1}\\gamma_{1}+\\cdots+l_{n}\\gamma_{n}+q_{f}(v))$$\n modulo terms of degree $\\geq3$ in $t$. So we need to resolve\n equations $a_{1}\\gamma_{1}+\\cdots+a_{n}\\gamma_{n}+a_{n+1}=0$ in variables\n $\\gamma_{1},\\ldots,\\gamma_{n}$ for all possible\n $(a_{1},\\ldots,a_{n+1})\\in W$.\n\n Consider the scalar product $k^{n+1}\\times k^{n+1}\\rightarrow k$,\n which sends $a\\times b$ to $a\\cdot b:=\n \\sum^{n+1}_{i=1}a_{i}b_{i}$.\n Using this scalar product, we will find the vector\n $(\\gamma_{1},\\ldots,\\gamma_{n},1)\\in W^{\\perp}:=\n \\{b\\in k^{n+1}:\\,b\\cdot a=0\\text{ for all } a\\in W\\}$.\n If $W\\subseteq k^{n}\\times\\{0\\}$, then we can put $\\gamma=(0,\\ldots,0)$\n to obtain the required curve.\n Otherwise, consider the  projection\n $\\pi:\\,k^{n+1}\\rightarrow k^{n}\\times\\{0\\}$, which assigns\n $(a_{1},\\ldots,a_{n},0)$\n  to\n $(a_{1},\\ldots,a_{n+1})$.\n And, let $\\tilde{\\pi}:\\,W\\rightarrow k^{n}\\times\\{0\\}$ be induced by $\\pi$.\n The projection  $\\tilde{\\pi}$ is injective, because $v\\in TC_{p}X$ and\n $l_{f}=0$ imply $q_{f}(v)=0$.\n If we denote $\\widetilde{W}=\\tilde{\\pi}(W)$, then\n $\\dim W^{\\perp}=\\dim\\widetilde{W}^{\\perp}$, by injectivity of $\\tilde{\\pi}$.\n Now, if $W^{\\perp}$ is of the form $W_{1}\\times\\{0\\}$,\n then by construction $\\widetilde{W}^{\\perp}=W_{1}\\times k$,\n contradicting with the equality of the dimensions.\n Therefore, there exists a vector\n $(\\gamma_{1},\\ldots,\\gamma_{n},1)\\in W^{\\perp}$,\n i.e., $l_{f}(\\gamma)+q_{f}(v)=0$ for all $f\\in I(X)$.\n We claim that this $\\gamma$ is linearly independent of $v$.\n Indeed, since $W$ is not included into $k^{n}\\times\\{0\\}$, we get\n $q_{f}(v)\\ne0$\n  for\n some $f\\in\\text{\\bf I}(X)$, which implies $l_{f}(\\gamma)\\ne0$.\n On the other hand, since $v\\in T_{p}X$, we get $l_{f}(v)=0$. Thus, we have found\n the desired $\\gamma$.\n \\end{pf}\n\n \\begin{rem}\n This result cannot be improved in the following sense. For the curve $X$,\n given by the equation $x^{2}=y^{3}$,\n in the affine plane, and $p=(0,0)$\n there is no smooth curve $C$ such that $I(p,C,X)\\geq4$.\n \\end{rem}\n\n \\begin{rem}\n In the case of analytic varieties the theorem is also valid.\n \\end{rem}\n\n\n\n \\section{An application to the affine schemes of algebras.}\\label{s:ap}\n\nThis section  shows that at least theoretically one may still be\nable to answer the\n problem of Shafarevich discussed in the introduction by solving quadratic equations arising from\n the second order obstructions to deformations of algebras.\n In particular, we find that the vectors, contributing to the discrepancy\n between the tangent spaces to the scheme of associative multiplications and the\n scheme\n of the degree 3 nilpotent multiplications, can be easily ``killed'' by the\n obstructions in many cases.  Then Theorem~\\ref{t:m} implies that  the tangent cones\n of the reduced schemes are the same, which means that the irreducible\n component of one variety is the component of the other one.\n\n Let us recall the notation from \\cite{sh,am}.\n The affine scheme $C_n$ of all multiplications on a fixed $n$-dimensional vector\n space $V$\n over a field $K$ (${\\rm char} K\\ne2$),\n which represent associative commutative algebras,\n is given by the equations of commutativity and associativity:\n $$c_{ij}^k=c_{ji}^k,\\qquad \\sum_{s=1}^n c_{ij}^s c_{sk}^l=\\sum_{s=1}^n c_{is}^l\n c_{jk}^s$$\n in the structure constants of multiplication\n $e_ie_j=\\sum_{k=1}^n c_{ij}^k e_k$\n of the basis $\\{e_1,\\ldots,e_n\\}$.\n Similar equations determine the affine scheme $A_n$ of commutative  nilpotent\n degree 3\n multiplications.  The reduced schemes associated to the above schemes are denoted\n $C_n^{\\rm red}$ and $A_n^{\\rm red}$, respectively.\n From \\cite{sh}, the irreducible components of $A_n^{\\rm red}$ have a very simple\n description\n $$A_{n,r}=\\{N\\in A_n^{\\rm red} |\\, \\dim N^2\\le r\\le \\dim {\\rm Ann}_N N\\},$$\n $1\\le r\\le(n-1)(n-r+1)\/2$, where $N$ denotes the algebra represented by the\n corresponding\n multiplication, $N^2$ is the square of the algebra and $\\rm Ann$ is the\n annihilator.\n\n Let $d:=n-r$, then, for $r=1,2$ and $r>(d^2-1)\/3$, Shafarevich in \\cite{sh} showed that\n $A_{n,r}$ is not a component of\n $C_n^{\\rm red}$, by constructing\n a line, contained in  $C_n^{\\rm red}$ but not in $A_n^{\\rm red}$, through a point of\n $A_{n,r}$.\n For other $r$, one has to compare\n the tangent spaces to the non-reduced schemes $A_n$ and $C_n$.\n The smooth set of $A_n^{\\rm red}$ is the union of\n $$U_{n,r}=\\{N\\in A_n^{\\rm red} |\\, \\dim N^2= r= \\dim {\\rm Ann}_N N\\}.$$\n If $W\\subset V$ is a subspace of dimension $r$, then\n the space $S_{n,r}=L(S^2(V\/W),W)$ of all linear maps from the symmetric product of\n $V\/W$ to $W$\n is naturally included as an affine subspace into  $A_{n,r}$. The group $G={\\rm\n GL}(V)$ acts\n on $C_n$ and $G S_{n,r}=A_{n,r}$.\n Then, the tangent space to $A_{n,r}$ at a point $N\\in S_{n,r}$ is\n $$T_N A_{n,r}=L(S^2(N\/N^2),N^2)+T_N G N,$$\n where the tangent space $T_N G N$ to the orbit is the space of\n maps from $L(S^2N,N)$ given by the coboundaries (in a Hochschild complex,\n see \\cite{am}) $x\\circ y= x\\varphi(y)-\\varphi(xy)+y\\varphi(x)$\n for some linear map $\\varphi:N@>>>N$.\n\n The tangent space $T_N C_n$\n always includes  $T_N A_{n,r}+F$, where\n the space $F$ consists of $x\\circ y=xf(y)+yf(x)$ for some $f\\in L(N\/N^2,K)\\subset\n L(N,K)$.\n To explain this difference\n Shafarevich embedded the scheme $C_n$ into the scheme $\\widetilde{C}_n$\n of commutative associative multiplications on a $(n+1)$-dimensional space which\n represent algebras with a unit $e$:\n $$C_n\\hookrightarrow\\widetilde{C}_n, \\qquad N\\mapsto N\\oplus Ke.$$\n In this situation, the subgroup $\\widetilde G$ of $GL(V\\oplus Ke)$ which fixes the\n unit\n acts on $\\widetilde{C}_n$. It turns out that\n $T_N GN+F$ is the tangent space to\n $\\widetilde{G} S_{n,r}$, whence  the equality\n \\begin{equation}\\label{e:tan}\n T_N \\widetilde{C}_n=T_N A_{n,r}+F\n \\end{equation}\n was enough to conclude that $A_{n,r}$ is the component of $C_n^{\\rm red}$.\n The equality was shown in \\cite{sh} for $3\\le r\\le (d+1)(d+2)\/6$,\n and this also holds $r=(5d^2-8d)\/16$ and $d$ divisible by 4\n by \\cite[Section~2.1]{am}.\n\n The original Shafarevich's method is very special to this situation and\n is  impractical\n to use it in general to explain the nilpotents in the structure sheaf of a scheme.\n We expect that the equality (\\ref{e:tan}) holds for all $3\\le r<(d^2-1)\/3$,\n which would leave only one case $r=(d^2-1)\/3$ unsettled. However, in this last case\n the tangent space to $C_n$ at a generic point in $A_{n,r}$ is too big:\n \\begin{equation}\\label{e:for}\n T_N C_n= L(S^2(N\/N^2),N^2)+T_N G N+L(S^2N_1,N_1),\n \\end{equation}\n where $N=N_1\\oplus N^2$ is a fixed decomposition.\n To show this one have to use Lemma~1 in \\cite{am}:\n a  generic\n algebra $N\\in A_{n,r}$, for $r\\le(d^2-1)\/3$, can be given by $d$ generators and a\n $(d(d+1)\/2-r)$-dimensional space  of  homogeneous degree 2 relations among the\n generators,\n so that\n the nilpotence of the third degree follows from this relations.\n But, for $r=(d^2-1)\/3$, this condition implies that there are no nontrivial\n relations among the\n homogeneous degree 2 relations, i.e., all of the relations $\\sum_i n_iz_i=0$ in\n $S^3N_1$,\n $n_i\\in N_1$, $z_i\\in Z:=ker(\\mu:S^2N_1@>>>N^2)$ ($\\mu$ is the multiplication\n on $N$),  are induced by a trivial in $N_1\\otimes Z$ element\n $\\sum_i n_i\\otimes z_i$. Indeed, $\\dim S^3N_1=d(d+1)(d+2)\/6$, while the number of\n equations\n that we get from the homogeneous degree 2 relations is equal to\n $$\\dim N_1\\otimes Z=d(d(d+1)\/2-(d^2-1)\/3)=d(d+1)(d+2)\/6.$$\n So, if there is a nontrivial relation among the\n homogeneous degree 2 relations, there would be not enough equations to deduce\n $N^3=0$.\n Since the only restriction in \\cite[Lemma~1]{am} for a vector from\n $L(S^2N_1,N_1)\\subset L(S^2N,N)$ to be tangent to the scheme $C_n$\n was arising from the nontrivial relations,\n we conclude (\\ref{e:for}).\n\n Anan'in suggested in \\cite{am} to use the second order obstructions to deformations\n of algebras\n to eliminate the excessive space   $L(S^2N_1,N_1)$. To be precise, suppose that\n $N\\in A_{n,r}$\n is a smooth point of $C_n^{\\rm red}$ and $\\circ\\in L(S^2N_1,N_1)$. Then, if $\\circ$ is\n tangent to $C_n^{\\rm red}$, the sum $\\circ+L(S^2(N\/N^2),N^2)$ also lies in the tangent\n space\n $T_N C_n^{\\rm red}$. A deformation of a commutative algebra on a vector space $V$\n with multiplication $x\\cdot y$ can be thought as\n a smooth curve $$x\\cdot y+(x\\circ y)t+(x\\star y)t^2+\\cdots$$ in\n the affine space $L(S^2V,V)$, where $t$ is a local parameter.\n It is a well know fact in algebraic geometry that\n through a smooth point $p$ of a variety $X$ we can  find\n a smooth curve inside $X$, whose tangent vector at $p$ is a given one in $T_p X$.\n Therefore, the smooth curves in $C_n^{\\rm red}$ with tangent vectors\n $\\circ+L(S^2(N\/N^2),N^2)$\n   give a lot of equations (the second order obstructions) arising from the\n associativity of\n multiplication:\n $$(x\\tilde\\circ y)\\tilde\\circ z-x\\tilde\\circ(y\\tilde\\circ z)=\n x(y\\star z)-(xy)\\star z+x\\star(yz)-(x\\star y) z$$\n for some $\\star\\in L(S^2N,N)$, where $\\tilde\\circ\\in\\circ+L(S^2(N\/N^2),N^2)$.\n A nice thing about these quadratic equations on $\\circ$ is that one can linearize\n them:\n $$(x\\circ y)* z+(x*y)\\circ z-x\\circ(y* z)-x*(y\\circ z)=\n x(y\\star z)-(xy)\\star z+x\\star(yz)-(x\\star y) z$$\n for all $*\\in L(S^2(N\/N^2),N^2)$.\n These equations have been used to prove Theorem~1 in \\cite{am} which implies that\n $A_{n,r}$ is the component of $C_n^{\\rm red}$ for $d(d+1)\/9\\le r\\le[d\/3](d-3)$\n (almost all cases that are not covered by \\cite{sh})\n if a certain algebra $N\\in A_{n,r}$ is a smooth point\n of $C_n^{\\rm red}$.\n Unfortunately, it seems impossible to prove that $N$ is the smooth point unless we\n know the tangent space at $N$.\n\n Now, we can apply Section~\\ref{s:pc}.\n Let $N\\in A_{n,r}$ with multiplication $x\\cdot y$ and $\\circ\\in TC_N C_n^{\\rm red}$.\n By Theorem~\\ref{t:m}, there exists a smooth plane curve $C$\n $$x\\cdot y+(x\\circ y)t+(x\\star y)t^2$$\n through $N$\n in the affine space $L(S^2V,V)$ of commutative\n multiplications on the vector space $V$, such that\n $\\circ$ is the tangent vector to the curve at $N$ and the multiplicity of\n intersection $I(p,C,C_n^{\\rm red})\\ge3$. Hence, the associativity equations in the\n structure\n constants imply\n \\begin{equation}\\label{e:ob}\n (x\\circ y)\\circ z-x\\circ(y\\circ z)=\n x(y\\star z)-(xy)\\star z+x\\star(yz)-(x\\star y) z\n \\end{equation}\n for some $\\star\\in L(S^2N,N)$. Note that we do not assume the smoothness condition\n at $N$.\n\n We want to deduce  the effective part of the obstructions to $\\circ$.\n As in \\cite[\\S1]{am}, decompose $\\circ$ and $\\star$ into the sum of linear maps\n $f_{ij}^k$ and $g_{ij}^k$ in $L(N_i\\otimes N_j,N_k)$, respectively,\n where $1\\le i,j,k\\le2$ and $N=N_1\\oplus N_2$,\n $N_2:=N^2$.\n For a sufficiently generic $N\\in A_{n,r}$, besides commutativity,\n $f_{ij}^k$\n  satisfy\n $$f_{12}^1=f_{22}^1=f_{22}^2=0,$$\n \\begin{equation}\\label{e:co}\n xf_{11}^1(y,z)+f_{12}^2(x,yz)=f_{12}^2(z,xy)+f_{11}^1(x,y)z,\n \\end{equation}\n by \\cite[\\S1]{am}. Moreover,\n $f_{12}^2$ satisfying (\\ref{e:co}) is unique for\n a given $f_{11}^1$, while its existence condition is described in\n \\cite[Lemma~1]{am}.\n The missing $f_{11}^2\\in L(S^2(N\/N^2),N^2)$ is always tangent\n to $C_n$.\n\n From (\\ref{e:ob}) we get\n \\begin{equation}\\label{e:ob1}\n f_{11}^1(f_{11}^1(x,y),z)-f_{11}^1(x,f_{11}^1(y,z))=\n -g_{12}^1(z,xy)+g_{12}^1(x,yz),\n \\end{equation}\n \\begin{equation}\\label{e:ob2}\n f_{12}^2(x,f_{12}^2(y,zt))-f_{12}^2(y,f_{12}^2(x,zt))=\n xg_{12}^1(y,zt)-yg_{12}^1(x,zt)\n \\end{equation}\n and\n $$f_{12}^2(f_{11}^1(x,y),zt)-f_{12}^2(x,f_{12}^2(y,zt))=xg_{12}^1(y,zt)-\n g_{22}^2(xy,zt),$$\n where $x,y,z,t\\in N_1$.\n The last\n equation determines $g_{22}^2$, and, one can check that\n commutativity of $g_{22}^2$ (surprisingly)\n follows  from  (\\ref{e:co}),\n (\\ref{e:ob1}) and (\\ref{e:ob2}).\n So, the only restrictions for $\\circ$ to lie in the tangent cone to $C_n$\n are (\\ref{e:co}),\n (\\ref{e:ob1}) and (\\ref{e:ob2}). The same argument as in \\cite[\\S1]{am}\n shows that $g_{12}^1$, satisfying (\\ref{e:ob1}), is unique  for a given\n $f_{11}^1$, and the existence condition is similar to \\cite[Lemma~1]{am}.\nWe have the following result:\n\n\\begin{thm}\\label{t:m1}\nThe variety $A_{n,r}$ is an irreducible component\n of $C_n^{\\rm red}$ when $r\\le(d^2-1)\/3$ if and only if,\n for some sufficiently generic $N\\in A_{n,r}$,\na solution $\\circ\\in L(S^2N,N)$  to (\\ref{e:co}),\n (\\ref{e:ob1}) and (\\ref{e:ob2}) has its part\n$f^1_{11}=0$ on the kernel of the multiplication\n$\\mu:S^2N_1@>>>N^2$ of the algebra $N$.\n\\end{thm}\n\n\\begin{pf}\nIf $A_{n,r}$ is an irreducible component\n of $C_n^{\\rm red}$, then, for some sufficiently generic $N\\in A_{n,r}$,\n the tangent cone to $C_n^{\\rm red}$ at $N$ coincides with\n$$T_N A_{n,r}=L(S^2(N\/N^2),N^2)+T_N G N.$$ But we know that the\nsolution to (\\ref{e:co}),\n (\\ref{e:ob1}) and (\\ref{e:ob2}) gives rise to a vector $\\circ$ from the\n tangent cone. Hence, $f^1_{11}=0$ on the kernel of the multiplication\n$\\mu:S^2N_1@>>>N^2$, because the vectors of $T_N G N$ are of the\nform $x\\circ y= x\\varphi(y)-\\varphi(xy)+y\\varphi(x)$\n for some linear map $\\varphi:N@>>>N$, while the vectors from\n $L(S^2(N\/N^2),N^2)$ clearly satisfy the property.\n\n Conversely, suppose that a vector $\\circ\\in L(S^2N,N)$ belongs to the tangent\n cone to the scheme $C_n$ at $N$. So, by the discussion above, it\n satisfies\n  (\\ref{e:co}),\n (\\ref{e:ob1}) and (\\ref{e:ob2}). Hence,  $\\circ$ has\n$f^1_{11}=0$ on the kernel of the multiplication of some\nsufficiently generic $N\\in A_{n,r}$. Then $f_{11}^1$ with\n$f_{12}^2(x,yz)=-xf_{11}^1(y,z)$ and\n$g_{12}^1(x,yz)=-f_{11}^1(x,f_{11}^1(y,z))$ are unique solutions\nto (\\ref{e:co}), (\\ref{e:ob1}) and (\\ref{e:ob2}). This shows that\n$\\circ$ is actually from $T_N A_{n,r}$.\n\\end{pf}\n\n\n We will finish this section showing the result of\n \\cite[Theorem~2]{am} with application of the above theorem and\n without the use of  Shafarevich's embedding $C_n\\hookrightarrow\\widetilde{C}_n$.\n\n\\begin{cor}\n$A_{n,r}$ is a  component\n of $C_n^{\\rm red}$ for the values $r=(5d^2-8d)\/16$, $d$ is divisible by 4.\n\\end{cor}\n\n\\begin{pf}\n In \\cite[Section~2.1]{am}, it was already shown that\n $T_N{C}_n=T_N A_{n,r}+F$ for a sufficiently generic algebra $N$.\n This implies that the part $f_{11}^1$ of a vector from $T_N{C}_n$ is determined by\n $f(x)y+f(y)x$, for some $f\\in L(N\/N^2,K)$, on the kernel of the multiplication\n $\\mu$ on $N$.\n The algebra $N$ had the property that the square of all $d$ generators is zero and\n for each generator $u$ there was a distinct generator $v$\n such that their product $uv$ also vanishes in $N$.\n Taking $x=y=u$ and $z=v$ in (\\ref{e:ob1}), we have\n $$f_{11}^1(2f(u)u,v)-f_{11}^1(u,f(u)v+f(v)u)=0.$$\n Since $uv$ and $uu$ are in the kernel of multiplication $\\mu$, we further get\n $$2f(u)(f(u)v+f(v)u)-f(u)(f(u)v+f(v)u)-f(v)(2f(u)u)=0,$$\n whence $f(u)^2v-f(u)f(v)u=0$. Therefore, $f(u)=0$ for all generators, and\n $f_{11}^1=0$\n on the kernel of $\\mu$ as it was required in Theorem~\\ref{t:m1}.\n\\end{pf}\n\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Appendix A}\\label{app:non_canonical}\n\nWe show, through an example,\nthat on the level of data languages we cannot hope for unique minimal\nQDA. Consider the QDA in Figure~\\ref{fig:even-sorted-QDA} over $PV =\n\\emptyset$ and $Y = \\{y_1,y_2\\}$.  It accepts all valuation words in\nwhich $d(y_1) \\le d(y_2)$ if $y_1$ is before $y_2$ and $y_1,y_2$ are\n  both on even positions, and all valuation words in which $y_2 < y_1$\n  or at least one of $y_1,y_2$ is not on an even position.\nHence, the data language define by this QDA consist of all data words\nsuch that the data on the even list positions is sorted. Since the QDA\nhas to check that each variable occurs exactly once, the number of states\nis minimal for defining this data language.\n\nHowever, the same data language would also be defined if the\n$\\blank$-transition from $q_3$ would be redirected to $q_5$. Then the\nsortedness property would only be checked for all $y_1,y_2$ with $y_2\n= y_1+2$, which is sufficient. This shows that the transition\nstructure of a state minimal QDA for a given data language is not\nunique.\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[node distance=40]\n\\node[state](q0){$q_0$};\n\\node[state,right of=q0](q5){$q_5$};\n\\node[state,accepting,right of=q5](q6){$q_6$};\n\\node[state,left of=q0](q7){$q_7$};\n\\node[state,below of=q0](q1){$q_1$};\n\\node[state, right of=q1](q2){$q_2$};\n\\node[state,right of=q2](q3){$q_3$};\n\\node[state,accepting, right of=q3](q4){$q_4$};\n\\node[state,right of=q2](q3){$q_3$};\n\\path (q4) node[right,xshift=10]{$\\scriptstyle d(y_1) \\le d(y_2)$}\n -- (q6) node[right,xshift=10]{\\scriptsize \\emph{true}}\n;\n\\draw[<-, shorten <=1pt] (q0.north) -- +(0, .3);\n\\draw[->]\n(q0) edge[bend left]  node{$\\scriptstyle \\blank$} (q1)\n     edge             node[above]{$\\scriptstyle (b,y_1)$} (q5)\n     edge node[swap]{$\\scriptstyle (b,y_2)$} (q7)\n(q1) edge  node[left]{$\\scriptstyle \\blank$} (q0)\n     edge             node[above]{$\\scriptstyle (b,y_1)$} (q2)\n\t   edge             node[left, very near start]{$\\scriptstyle (b,y_2)$} (q7)\n(q2) edge  node[above]{$\\scriptstyle \\blank$} (q3)\n     edge  node[very near end, yshift=-7] {$\\scriptstyle (b, y_2)$} (q6)\n(q3) edge[bend left]  node[below]{$\\scriptstyle \\blank$} (q2)\n     edge             node[above]{$\\scriptstyle (b,y_2)$} (q4)\n(q4) edge[loop above] node{$\\scriptstyle \\blank$} ()\n(q5) edge[loop below] node{$\\scriptstyle \\blank$} ()\n     edge             node[above]{$\\scriptstyle (b,y_2)$} (q6)\n(q6) edge[loop below] node{$\\scriptstyle \\blank$} ()\n(q7) edge[loop below] node{$\\scriptstyle \\blank$} ()\n     edge[bend left=40]             node[very near start, above, yshift=5]{$\\scriptstyle (b,y_1)$} (q6)\n;\n\\end{tikzpicture}\n\\caption{A QDA expressing the property that the data on the even\n  positions in the list is sorted.} \\label{fig:even-sorted-QDA}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Appendix B}\n\nAngluin \\cite{DBLP:journals\/iandc\/Angluin87} introduced a popular learning framework, which is originally designed to learn regular languages. In this framework, a \\emph{learner} (or learning algorithm) learns a regular language $L$, the so-called \\emph{target language}, over an a priory fixed alphabet $\\Sigma$ by actively querying a \\emph{teacher}. The teacher is capable of answering two different kinds of queries: \\emph{membership} and \\emph{equivalence queries}. On a membership query, the learner presents a word $w \\in \\Sigma^\\ast$, and the teacher replies ``yes'' or ``no'' depending on whether $w$ belongs to $L$ or not. On an equivalence query, the learner conjectures a regular language $H \\subseteq \\Sigma^\\ast$, typically given as a finite automaton, and the teacher checks whether $H$ is an equivalent description of $L$. If this is the case, he replies ``yes''. Otherwise, he returns a \\emph{counter-example} $w \\in L \\Leftrightarrow w \\not\\in H$.\n\nIn \\cite{DBLP:journals\/iandc\/Angluin87}, Angluin presented a learning algorithm that learns a regular language in time polynomial in the size of the (unique) minimal deterministic finite automaton accepting the target language and the length of the longest counter-example returned by the teacher. Angluin's algorithm maintains a prefix-closed set $S \\subseteq \\Sigma^\\ast$, a suffix-closed set $E \\subseteq \\Sigma^\\ast$, and stores the learned data in a table (realized as a mapping $T\\colon (S \\cup S\\Sigma) E \\to \\{0,1\\}$), whose rows are labeled with strings from $S$ and whose columns are labeled with string from $E$. The key idea of the algorithm is to approximate the Nerode congruence of the target language using strings from $S$ as representatives for the equivalence classes and strings from $E$ as samples to distinguish these classes. New strings are added to $S$ and $E$ whenever necessary until an equivalence query reveals that the conjectured automaton is equivalent to the target language. \n\nAlthough originally introduced to learn regular languages, this algorithm can be easily lifted to the learning of Moore machines. In this setting, the ``target language'' is a finite-state computable mapping $\\lambda \\colon \\Sigma^\\ast \\to \\Gamma$ (i.e., a mapping computable by a Moore machine) that maps each word $w \\in \\Sigma^\\ast$ to some output $\\lambda(w)$ taken from a finite set $\\Gamma$ of output symbols. (We obtain Angluin's setting for $\\Gamma = \\{0,1\\}$.) Moreover, membership queries ask now for the output---or classification---of a word rather then whether it belongs to a language or not. Finally, on an equivalence query, the learner proposes a Moore machine $\\mathcal M$. If $\\mathcal M$ is not equivalent to the target language, the teacher returns a counter-example $w$ such that the output of $\\mathcal M$ on $w$ is different from $\\lambda(w)$.\n\n\nAdapting Angluin's algorithm to work with Moore machines is straightforward. Since the Nerode congruence can easily be lifted to the Moore machine setting, it is indeed enough to change the table to a mapping $T\\colon (S \\cup S\\Sigma) E \\to \\Gamma$; everything else can be left unchanged.  \nThis adapted algorithm also learns the minimal Moore machine for the target language in time polynomial in this minimal Moore machine and the length of the longest counter-example returned by the teacher.\n\n\n\\section*{Appendix C}\n\n\nIn this appendix we describe, in a greater detail, the translation from an EQDA $\\mathcal{A}$ to a formula $\\mathcal{T}(\\mathcal{A})$ expressed in $\\Strand$ or the APF such that the set of data words accepted by $\\mathcal{A}$ corresponds to the program configurations $\\mathcal{C}$ which model $\\mathcal{T}(\\mathcal{A})$.\n\nRecall the formal definition of an EQDA from Section~\\ref{sec:elastic}. In an EQDA $\\mathcal{A} = (Q, q_0, \\Pi, \\delta, f)$ over program variables $PV$ and universal variables $Y$, each transition on $\\blank$ is a self loop. Without restricting the class of languages accepted by $\\mathcal{A}$ we assume, for the purpose of translation, that our EQDAs have three additional properties.\n\nFirstly, we assume that any path in the EQDA, along which a universal variable occurs together with auxiliary variables like $nil$ which are introduced by the encoding from Section~\\ref{sec:modeling}, goes to the formula $true$. This does not change the language accepted by the automaton as it still accepts all data words respecting the formula constraints for other valuations of the universal variables and where the data at these auxiliary variables can be any data value.\n\n\\begin{figure}\n\\vspace{-0.4cm}\n\t\\centering\n\t\\subfigure[]{\n\t\t\\begin{tikzpicture}\n\t\t\t\\node[state]             (1) {$q_1$};\n\t\t\t\\node[state, right of=1] (2) {$q_2$};\n\t\t\t\\draw[<-] (1.west) -- +(-.5, 0);\n\t\t\t\\draw[->] (2.east) -- +(.5, 0);\n\t\t\t\\draw[->] (1) edge             node {\\scriptsize $(b, y)$} (2)\n\t\t                   (2) edge[loop above] node {\\scriptsize $\\blank$} ();\n\t\t\\end{tikzpicture}\n\t\t\\label{fig:relevant-loops-a}\n\t}\n\t%\n\t\\hskip 3em\n\t%\n\t\\subfigure[]{\n\t\t\\begin{tikzpicture}\n\t\t\t\\node[state]             (1) {$q_1$};\n\t\t\t\\node[state, right of=1] (2) {$q_2$};\n\t\t\t\\draw[<-] (1.west) -- +(-.5, 0);\n\t\t\t\\draw[->] (2.east) -- +(.5, 0);\n\t\t\t\\draw[->] (1) edge             node {\\scriptsize $(b, y)$} (2)\n\t\t                       edge[loop above] node {\\scriptsize $\\blank$} ();\n\t\t\\end{tikzpicture}\n\t\t\\label{fig:relevant-loops-b}\n\t}\n\\caption {Base cases for detecting irrelevant self-loops.}\n\\vspace{-0.3cm}\n\\end{figure}\n\nSecondly, we assume that the EQDA has no \\emph{irrelevant} loops which are defined inductively as follows:\nfix a simple path $p$ of an EQDA $\\A$ that leads from the initial to an\naccepting state, and on $p$ consider a state $q_1$ (see\nFig~\\ref{fig:relevant-loops-a}) which reads a universal variable $(b,\ny)$ on a transition to state $q_2$. If $q_2$ has a self loop on the\nblank symbol, i.e. $\\delta(q_2, \\underline{b}) = q_2$, then this loop\nis inductively defined to be \\emph{irrelevant on $p$} if $q_1$ has no\nself-loop, or if the self-loop at $q_1$ is also irrelevant on\n$p$. Symmetrically, a self-loop at $q_1$ is irrelevant on $p$ if $q_2$\nhas no self-loop or has one which is irrelevant on $p$ (see\nFig~\\ref{fig:relevant-loops-b}).  If a self-loop is irrelevant on $p$,\nthen it can be omitted for words accepted along the path $p$.  To see\nwhy, consider a valuation word $v = \\dots\n(b, y)~ \\blank\\dots$ that is accepted along $p$ using the self-loop in\nFigure~\\ref{fig:relevant-loops-a}.  A different valuation $v' = \\dots \\blank~(b, y)\\dots$ of the\nsame data word is rejected by $\\mathcal{A}$\nsince $q_1$ has no transition on $\\underline{b}$. Hence, the data word\ncorresponding to $v$ is not accepted by $\\A$.\nWe can remove irrelevant loops from $\\A$ without changing the accepted\ndata language by simply removing those loops that are irrelevant on\neach path they occur on, or by splitting states if they have a\nself-loop that is irrelevant only on some paths.\n\n\nThirdly, we assume that the universal variables are read by the EQDA in a particular order and all paths in the EQDA that do not respect this order lead to the formula $\\true$.\nThe translation that we give below considers each path of the automaton separately. Thus, if the automaton does not satisfy the above property, then for any path that does not read the variables in the correct order we rename the variables on the transitions and in the data formulas along that path accordingly before the translation.\n\nLet us now turn to the translation of the paths. We observe that all variables appear exactly once in any valuation word accepted by $\\mathcal{A}$. Since we disallow universal variables to appear together, this is ensured by adding some dummy symbols where these variables can appear in case the valuation word is too short. A consequence of this property is that there can be no cycle in our EQDA model which shares an edge labeled with a (universal or pointer) variable. Consider a simple path $p$ of the automaton from the initial state $q_0$ to the output state $q_p$,\n$q_0 \\xrightarrow{\\pi_0} \\ldots \\xrightarrow{\\pi_{n-1}} q_p$ ($\\pi_i \\in \\Pi \\neq \\underline{b}$). Below we informally describe the translation $\\mathcal{T}$  from path $p$ to a formula $\\phi_p$ which captures the relative positions of the pointer and universal variables along $p$ and forms the guard of a universally quantified implication in a conjunct of the translated formula.\nAt a higher level, whenever a state $q$  in path $p$ has a self-loop on the blank symbol $\\underline{b}$, pointers and universal variables $v_1, v_2 \\in PV \\cup Y$ labeled along the incoming and outgoing transitions of this state are constrained by the relation $v_1 < v_2$ or $v_1 \\rightarrow^+ v_2$. The presence of a self loop ensures that the variables are related by an elastic relation which is required for decidability in \\Strand and APF. On the other hand, if  $q$ has no transition on $\\blank$, then the pointers labeled along the incoming and outgoing transitions are constrained by the successor relation. Note that successor is an inelastic relation and is not allowed to relate two universal variables. In this case we identify a state $q'$ on path $p$, closest to $q$, which has a transition on some pointer (non-universal) variable $pv$. Since we have already stripped our EQDA of all \\emph{irrelevant } loops, the subpath from $q'$ to $q$ has no self-loops. Thus, the universal variables at $q$ can be constrained to be a fixed distance away from the pointer $pv$. This is allowed in APF using arithmetic on the pointer variables. For \\Strand, the same effect can be obtained by introducing a monadic predicate which tracks the distance of the universal variable from the pointer variable $pv$.\n\nWe skip a formal description of the translation. A subtle case to note, however, is when a state $q$ in path $p$ has a self loop on the blank symbol $\\underline{b}$ and the incoming and outgoing transitions on $q$ are both labeled by letters of the form $(b, y)$ where $y \\in Y$. Unlike \\Strand, APF forbids two adjacent universal variables $y_1, y_2$ to be related by $<$. And so for the case of arrays, translation $\\mathcal{T}(p)$ constrains these universal variables as $y_1 \\leq y_2$.  Moreover, we modify the output of the final state\nalong this path $f_{\\mathcal{A}}(q_p)$ to include the data constraint $d(y_1) = d(y_2)$ if it was not already implied by the output formula. Note that at this point the constraint does not capture the exact semantics of the automaton.\n\n\nThe universally quantified formula that is captured by this particular path $p$ is\n$\\forall Y. ~\\phi_p \\Rightarrow f_{\\mathcal{A}}(q_p)$. We construct these\nimplications for all simple paths in the EQDA and conjunct them to get the\nfinal formula. All other paths in $\\mathcal{A}$ semantically go to\n$\\false$. Hence, we also add a conjunct $\\forall Y.~ \\neg (\\bigvee_p\n\\phi_p) \\Rightarrow \\mathit{false}$. So for an EQDA $\\mathcal{A}$,\n$\\mathcal{T}(\\mathcal{A}) = \\bigwedge_p \\forall Y.~ \\phi_p \\Rightarrow f_{\\mathcal{A}}(q_p) ~\\bigwedge~ \\forall Y.~\\neg (\\bigvee_p \\phi_p) \\Rightarrow \\textit{false}$.  \nSince negation is arbitrarily allowed over atomic formulas\nin \\Strand, $\\mathcal{T}(\\mathcal{A})$ is clearly in the decidable\nfragment of \\Strand. APF also allows negation over atomic formulas\nwhich relate two pointer variables or a universal variable with a\npointer variable.  However, negation of an atomic formula $y_1 \\leq\ny_2$ is not allowed in APF. But since we assume for the translation\nthat the automaton considers a fixed variable ordering on $Y$ and all\nother paths with a different ordering lead to $\\true$, we can simply\nremove negations of formulas $y_1 \\leq y_2$ from $\\neg \\bigvee_p\n\\phi_p$.\n\n\n\n\\section*{Appendix D}\n\nIn this appendix, we sketch how an active learning algorithm can be used to learn program invariants expressible in the array property fragment and the \\Strand decidable fragment over lists.\nInvariant synthesis \ncan be achieved using\n\\emph{two} distinct procedures: (a) building the learner according to the learning algorithm described in Section~\\ref{sec:learning}, and (b) building\na teacher which can answer questions about invariant for a particular program.\nAn acceptable invariant for a program, in general, has to satisfy \\emph{three} properties:\n it must include the pre-condition,\n it must be contained in the post-condition, and\n it must be inductive.\nMoreover, in order to certify the above indeed hold, the invariant should be expressible in a logic that permits\na decidable satisfiability problem for the above conditions.\n\nBuilding an adequate teacher is not easy as the\ninvariant is \\emph{unknown}, and the whole point of learning is to find the invariant.\nStill the teacher certainly has \\emph{some knowledge} about the set of structures\nin the invariant and can answer certain questions with certainty. For example, when asked whether a data word $w$ belongs\nto the invariant $I$, if $w$ belongs to the pre-condition (or the strongest post-condition of the pre-condition), \n she can definitely say that $w$ belongs to $I$.\nAlso, when $w$ belongs to the negation of the post-condition (or to the weakest pre-condition of the negated post-condition), the teacher can definitely answer that $w$ does not belong to $I$.\nFor other queries, in general, the teacher\ngives arbitrary answers\nand these answers determine the kind of invariant that is finally learned.\n Turning to equivalence queries, if the learned invariant falls within a decidable fragment (as is ensured by the above learning algorithm) and the pre\/post-condition and the program body is such that the verification conditions \nare expressed in appropriate decidable logics\n(\\Strand\/APF),\na teacher is able to check if the conjectured\ninvariant is adequate and satisfies the above three conditions.\nIf the invariant is inadequate and does \nnot include the pre-condition or intersects the negation of the postcondition, then the teacher can find an appropriate counterexample\nto report to the learner. If the inadequacy is due to the conjecture\nnot being inductive, then the teacher would find a pair of configurations\n$(c,c')$ such that $c$ is allowed by the conjecture\nwhile $c'$ is reachable from $c$ and is excluded from it,\nand decide\nto either report $c$ or $c'$ as a counterexample. This choice \nagain determines the final invariant being learned, similar to membership queries that the teacher is unsure about.\n\nThe idea is to pit such a teacher against a learner in order to learn the invariant, \\emph{despite} the fact that the teacher\ndoes not know the invariant herself. The learner's objective is to learn the \\emph{simplest} data automaton that captures the knowledge the teacher has.\nThe key property that the learner relies on is Occam's razor --- that the simplest set (i.e., the automaton with the least number of states) consistent with the queries answered by the teacher is a likely invariant. Note that the learner will not, in general, simply learn an automaton that captures precisely the knowledge of the inadequate teacher;\nrepresentation of this knowledge is often far more complex than a true invariant. In other words,\nthe learner will learn the simplest automaton that \\emph{generalizes} the partial knowledge the teacher has.\n\n\n\n\\newpage\n\\section*{Appendix E}\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[align=center,node distance=40]\n\\node[state](q0){$q_0$};\n\\node[state,below right=1.7cm and 1.0cm of q0](q1){$q_1$};\n\\node[state,below left=1.7cm and 1.0cm of q0](q2){$q_2$};\n\\node[state,below of=q2](q8){$q_8$};\n\\node[state,below left=1.7cm and 1.5cm of q8](q7){$q_7$};\n\\node[state,below right=2.2cm and 0.3cm of q1](q6){$q_6$};\n\\node[state,below left=1.3cm and 1.8cm of q6](q4){$q_4$};\n\\node[state,below left=1.7cm and 0.5cm of q4](q10){$q_{10}$};\n\\node[state,below right=1.7cm and 1.3cm of q4](q12){$q_{12}$};\n\\node[state,accepting,below left=1.0cm and 0.5cm of q10](q13){$q_{13}$};\n\\node[state,accepting,below right=1.0cm and 0.5cm of q12](q11){$q_{11}$};\n\\node[state,below right=1.8cm and 1.6cm of q6](q5){$q_{5}$};\n\\node[state,below right=1cm and 1.9cm of q5](q3){$q_3$};\n\\node[state,accepting,below of=q3](q9){$q_9$};\n\\node[state,above=4cm of q3](q14){$q_{14}$};\n\n\\path (q13) node[below,yshift=-10]{$\\scriptstyle d(y_1) \\le d(y_2) \\wedge$}\n -- (q13) node[below,yshift=-20]{$\\scriptstyle d(y_1) < k \\wedge d(y_2) < k$}\n -- (q11) node[below,yshift=-10]{$\\scriptstyle d(y_1) \\le d(y_2) \\wedge d(y_1) < k$}\n -- (q9) node[below,yshift=-15]{$\\scriptstyle d(y_1) \\le d(y_2)$}\n\n;\n\\draw[<-, shorten <=1pt] (q0.north) -- +(0, .3);\n\\draw[->]\n(q0) edge node{$\\scriptstyle (\\{nil\\}, \\yblank)$} (q1)\n(q0) edge node[left]{$\\scriptstyle (\\{cur,nil\\}, \\yblank)$} (q2)\n(q2) edge node[near start]{$\\scriptstyle (\\{head\\}, \\yblank)$} (q8)\n(q8) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q8) edge node[near end, yshift=4]{$\\scriptstyle (b, y_1)$} (q7)\n(q2) edge[bend right] node[left]{$\\scriptstyle (\\{head\\}, y_1)$} (q7)\n(q7) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q7) edge node[left]{$\\scriptstyle (b, y_2)$} (q13)\n(q13) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q1) edge[bend right] node[left, yshift=-17, xshift=-2]{$\\scriptstyle (\\{head\\}, y_1)$} (q4)\n(q4) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q4) edge node[left]{$\\scriptstyle (b, y_2)$} (q10)\n(q10) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q10) edge node[xshift=-3]{$\\scriptstyle (\\{cur\\}, \\yblank)$} (q13)\n(q12) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q11) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q12) edge node[left]{$\\scriptstyle (b, y_2)$} (q11)\n(q4) edge node[left, very near end, yshift=6, xshift=-3]{$\\scriptstyle (\\{cur\\}, \\yblank)$} (q12)\n(q4) edge[bend left] node[right]{$\\scriptstyle (\\{cur\\}, y_2)$} (q11)\n\n(q1) edge node[left, very near end, yshift=7]{$\\scriptstyle (\\{head\\}, \\yblank)$} (q6)\n(q6) edge node[left, near start]{$\\scriptstyle (b, y_1)$} (q4)\n(q6) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q6) edge[bend right] node[left, yshift=14, xshift=-12]{$\\scriptstyle (\\{cur\\}, y_1)$} (q3)\n(q3) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q5) edge node[xshift=-6]{$\\scriptstyle (b, y_1)$} (q3)\n(q5) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q6) edge node[very near start, yshift=-8]{$\\scriptstyle (\\{cur\\}, \\yblank)$} (q5)\n(q1) edge[bend left] node[left, yshift=3]{$\\scriptstyle (\\{head, cur\\}, y_1)$} (q3)\n\n(q3) edge node{$\\scriptstyle (b, y_2)$} (q9)\n(q9) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q14) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q1) edge node[above, near end, yshift=5, xshift=-2]{$\\scriptstyle (\\{head, cur\\}, \\yblank)$} (q14)\n(q14) edge node{$\\scriptstyle (b, y_1)$} (q3)\n;\n\\end{tikzpicture}\n\\caption{The EQDA expressing the invariant of the program which finds a key $k$ in a sorted list. Here \\emph{head} and \\emph{cur} are pointer variables and {\\emph k} is an integer variable in the program.}\n\\end{figure}\n\n\n\n\n\\section*{Appendix B}\n\n\n\n\\vspace{-0.1cm}\n\\begin{theorem}\n\\label{thm-canonical}\nFor each QDA $\\A$ there is a unique minimal QDA $\\A'$ that accepts the\nsame set of valuation words.\n\\end{theorem}\n\\vspace{-0.1cm}\n\\emph{Proof}\nConsider a language $\\Lval$ of valuation words that can be accepted by\na QDA, and let $w \\in \\Pi^*$ be a symbolic word. Then there must be a\nformula in the lattice that characterizes precisely the data\nextensions $v$ of $w$ such that $v$ in $\\Lval$. Since we assume that all\nthe formulas in the lattice are pairwise non-equivalent, this formula is uniquely determined. \nIn fact, take any QDA $\\A$ that accepts $\\Lval$. Then $w$ leads to some state $q$ in\n$\\A$ that outputs some formula $f(q)$. If $w$ leads to any other\nformula in another QDA $\\A'$, then $\\A'$ accepts a different language of\nvaluation words.\n\nThus, a language of valuation words can be seen as a function that\nassigns to each symbolic word a uniquely determined formula, and a QDA can be viewed as a\nMoore machine that\ncomputes this function. For each such Moore machine there exists a\nunique minimal one that computes the same function, hence the theorem.\n\\qed\n\n\nAs the proof above shows, we can view a language of valuation words as a function that\nmaps to each symbolic word a uniquely determined formula, and a QDA can be viewed as a\nMoore machine (an automaton with output function on states) that\ncomputes this function.\n\n\n\n\n\n\\section*{Appendix C}\n\n\n\n\nThe learning algorithm that we use to synthesize QDAs does not\nconstruct EQDAs in general. However, we can show the following surprising \nresult that every QDA $\\A$ can be \\emph{uniquely over-approximated}\nby a language of valuation words that can be\naccepted by an EQDA $\\AEL$. \nWe will refer to this construction, which we outline below, as \\emph{elastification}.\nThis construction crucially\nrelies on the particular structure that elastic automata have, which forces\na unique set of words to be added to the language in order to make it elastic.\n\nLet $\\A = (Q, q_0, \\Pi, \\delta, f)$ be a QDA and for a state $q$ let\n$R_{\\blank}(q):= \\{q' \\mid q \\xrightarrow{\\blank}^* q' \\}$\nbe the set of states reachable from $q$ by a (possibly empty) sequence\nof $\\blank$-transitions.  For a set $S \\subseteq Q$ we let\n$R_{\\blank}(S) := \\bigcup_{q \\in S} R_{\\blank}(q)$.\n\nThe set of states of $\\AEL$ consists of sets of states of $\\A$ that\nare reachable from the initial state $R_{\\blank}(q_0)$ of $\\AEL$ by\nthe following transition function (where $\\delta(S,a)$ denotes the\nstandard extension of the transition function of $\\A$ to sets of\nstates):\\\\\n$\\delta_{\\text{el}}(S,a) =\n\\begin{cases}\nR_{\\blank}(\\delta(S,a)) & \\mbox{if } a \\not= \\blank \\\\\nS  & \\mbox{if } a = \\blank \\mbox{ and $\\delta(q,\\blank)$ is defined  for some $q \\in S$} \\\\\n\\mbox{undefined}  & \\mbox{otherwise.} \\\\\n\\end{cases}\n$\\\\\nNote that this construction is similar to the usual powerset construction\nexcept that in each step we apply the transition function of $\\A$ to the\ncurrent set of states and take the $\\blank$-closure. However, if the\ninput letter is $\\blank$, $\\AEL$ loops on the current\nset if a $\\blank$-transition is defined for some state in the set.\n\nThe final evaluation formula for a set is the least upper bound of the\nformulas for the states in the set:\n$f_{\\text{el}}(S) = \\bigsqcup_{q \\in S}f(q)$. \nWe can now show that $\\AEL$  is the \n\\emph{most precise over-approximation} of the language of valuation words accepted by QDA $\\A$.\n\n\\vspace{-0.15cm}\n\\begin{theorem}\\label{thm:elastification}\nFor every QDA $\\A$, the EQDA $\\AEL$ satisfies \n$\\Lval(\\A) \\subseteq \\Lval(\\AEL)$, and \nfor every EQDA $\\B$ such that $\\Lval(\\A) \\subseteq \\Lval(\\B)$,\n $\\Lval(\\AEL) \\subseteq \\Lval(\\B)$ holds.\n\\end{theorem}\n\\emph{Proof:~}\nNote that $\\AEL$ is elastic by definition of $\\delta_{\\text{el}}$. It\nis also clear that $\\Lval(\\A) \\subseteq \\Lval(\\AEL)$ because for each\nrun of $\\A$ using states $q_0 \\cdots q_n$ the run of $\\AEL$ on the\nsame input uses sets $S_0 \\cdots S_n$ such that $q_i \\in S_i$, and by\ndefinition $f(q_n)$ implies $f_{\\text{el}}(S_n)$.\n\nNow let $\\B$ be an EQDA with $\\Lval(\\A) \\subseteq \\Lval(\\B)$. Let $w =\n(a_1,d_1) \\cdots (a_{n},d_{n})\\in \\Lval(\\AEL)$ and let $S$ be the\nstate of $\\AEL$ reached on $w$.  We want to show that $w \\in\n\\Lval(\\B)$.  Let $p$ be the state reached in $\\B$ on $w$. We show that\n$f(q)$ implies $f_\\B(p)$ for each $q \\in S$. From this we obtain\n$f_{\\text{el}}(S) \\Rightarrow f_\\B(p)$ because $f_{\\text{el}}(S)$ is\nthe least formula that is implied by all the $f(q)$ for $q \\in S$.\n\n\n\nPick some state $q \\in S$. By definition of $\\delta_{\\text{el}}$ we\ncan construct a valuation word $w'$ that leads to the state $q$ in\n$\\A$ and has the following property: if all letters of the form\n$(\\blank,d)$ are removed from $w$ and from $w'$, then the two\nremaining words are the same. In other words, $w$ and $w'$ can be\nobtained from each other by inserting and\/or removing\n$\\blank$-letters.\n\n\nSince $\\B$ is elastic, $w'$ also leads to $p$ in $\\B$. From this we\ncan conclude that $f(q) \\Rightarrow f(p)$ because otherwise there\nwould be a model of $f(q)$ that is not a model of $f(p)$ and by\nchanging the data values in $w'$ accordingly we could produce an input\nthat is accepted by $\\A$ and not by $\\B$.\n\\qed\n\n\\subsection*{A Teacher for Invariants over Linear Data Structures}\n\n\n\\noindent {\\bf Implementing the learner.}\nWe used the \\libalf library \\cite{DBLP:conf\/cav\/BolligKKLNP10} as an implementation of the active learning algorithm~\\cite{DBLP:journals\/iandc\/Angluin87}.\nWe adapted its implementation to our setting by modeling QDAs as Moore machines.\nIf the learned QDA is not elastic, we elastify it as described in Section~\\ref{sec:elastic}.\nThe result is then converted to a quantified formula over \\Strand or the APF\nand we check if the learned invariant was adequate using a constraint solver.\n\\begin{table*}[th]\n\t\\centering \\footnotesize\n\t\\vspace{-0.55cm}\n\n\n\n\n\n\n    \n     \n\n\n\n\n\n\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\t\n\\begin{tabular}{||l|| r| r||r|r|r|c|r||}\n\t\\hline\n\t~~~Example & \\#Test & $T_{teacher}$ & \\#Eq. & \\#Mem. & Size~~ & Elastification & $T_{learn}$ \\\\\n\t             &    inputs &   (s) &    &       & \\#states & required ?   & (s)\\\\\\hline\t\n\narray-find\t    &310\t& 0.05 &  2 & 121   & 8  & no  &  0.00 \\\\\\hline\narray-copy\t&7380 & 1.75   & 2  & 146   & 10 & no   & 0.00 \\\\\\hline\narray-comp\t&7380 & 0.51    & 2  & 146   & 10 & no  & 0.00 \\\\\\hline\nins-sort-outer  &363 & 0.19    & 3  & 305   & 11 & no   & 0.00 \\\\\\hline\nins-sort-innner &363& 0.30  & 7  & 2893  & 23 & yes  & 0.01 \\\\\\hline\nsel-sort-outer  &363& 0.18  & 3  & 306   & 11 & no   & 0.01 \\\\\\hline\nsel-sort-inner  &363& 0.55   & 9 &  6638  & 40 & yes  & 0.05 \\\\\\hline\\hline\n\n\nlist-find\t    &111& 0.04   & 6 & 1683   & 15 & yes & 0.01 \\\\\\hline\nlist-insert\t    &111& 0.04   & 3 & 1096   & 20 & no   & 0.01 \\\\\\hline\nlist-init\t    &310& 0.07   & 5 & 879    & 10 & yes  & 0.01 \\\\\\hline\nlist-max\t    &363& 0.08   & 7 & 1608   & 14 & yes  & 0.00 \\\\\\hline\nlist-merge\t    &5004& 10.50  & 7 & 5775   & 42 & no   & 0.06 \\\\\\hline\nlist-partition  &16395 & 11.40    & 10 & 11807 & 38 & yes  & 0.11 \\\\\\hline\nlist-reverse\t&27& 0.02    & 2 & 439    & 18 & no   & 0.00 \\\\\\hline\n\n\nlist-bubble-sort   &363& 0.19 & 3 & 447    & 12 & no   & 0.01 \\\\\\hline\nlist-fold-split\t  &1815& 0.21   & 2 & 287    & 14 & no  & 0.00 \\\\\\hline\nlist-quick-sort\t   &363 & 0.03 & 1 & 37     & 5  & no  & 0.00 \\\\\\hline\n\nlist-init-cmplx    &363& 0.05 & 1 & 57     & 6  & no   & 0.01 \\\\\\hline\\hline\n\nlookup\\_prev   &111& 0.04  & 3  & 1096  & 20  & no   & 0.01 \\\\\\hline\nadd\\_cachepage &716& 0.19  & 2  & 500   & 14  & no   & 0.01 \\\\\\hline\n\nsort (merge)\t\t&363& 0.04  & 1 & 37     & 5  & no   & 0.00 \\\\\\hline\ninsert\\_sorted\t\t&111& 0.04  & 2 & 530     & 15 & no   & 0.01 \\\\\\hline\n\ndevres\t\t   &372& 0.06 & 2  & 121   & 8   & no   & 0.00 \\\\\\hline\nrm\\_pkey\t\t   &372& 0.06 & 2  & 121   & 8   & no   & 0.00 \\\\\\hline\\hline\n\n\\multicolumn{8}{|c|}{Learning Function Pre-conditions}\\\\\\hline\nlist-find\t\t    &111& 0.01   & 1 & 37    & 5  & no   & 0.00 \\\\\\hline\nlist-init\t\t    &310& 0.02   & 1 & 26\t& 4  & no   & 0.00 \\\\\\hline\nlist-merge\t\t    &329& 0.06   & 3 & 683   & 19 & no   & 0.01 \\\\\\hline\n\n\t\\end{tabular}\n\t\\caption{Results of our experiments.\\label{tbl:experimental_results}}\n\\vspace{-1cm}\n\\end{table*}\n\n{\\bf Experimental Results.}\\footnote{Our prototype implementation along with the results for all our programs can be found at \\url{http:\/\/automata.rwth-aachen.de\/~neider\/learning_qda\/}}.\nWe evaluate our approach on a suite of programs  (see Table~\\ref{tbl:experimental_results}) for learning invariants and preconditions.\nFor every program, we report the\nthe number of test inputs and the time ($T_{teacher}$) taken to build the teacher from the samples collected along these test runs. We also report the number of equivalence and membership queries answered by the teacher in the active learning algorithm, the size of the final elastic automata, whether the learned QDA required any elastification and finally, the time ($T_{learn}$) taken to learn the QDA.\n\nThe array programs are programs for finding a key in an array, copying and comparing two arrays, and inner and outer loops of insertion and selection sort over an array.\nThe list programs find and insert a key in a sorted list, initialize a list, return the maximum data value in a list, merge two disjoint lists, partition a list into two lists depending on a predicate and reverse in-place a sorted list. The programs bubble-sort, fold-split and quick-sort are taken from~\\cite{celia}.\nThe program \\emph{list-init-cmplx}\nsorts an input array using heap-sort and then initializes a list with the contents of this sorted array.\nSince heap-sort is a complex algorithm that views an array as a binary tree,\nnone of the current automatic white-box techniques for invariant synthesis can handle such complex programs.\nHowever, our learning approach being black-box, we are able to learn the correct invariant, which is that the list is sorted. Similarly synthesizing post-condition annotations for recursive procedures like merge-sort and quick-sort is in general difficult for white-box techniques, like \\emph{interpolation}, which require a post-condition.\n\nThe methods \\emph{lookup\\_prev} and \\emph{add\\_cachepage} are from the module cachePage in a verified-for-security platform for mobile applications~\\cite{asplos13}. The module cachePage maintains a cache of the recently used disc pages as a priority queue based on a sorted list.\nThe method \\emph{sort} is a merge sort implementation and \\emph{insert\\_sorted} is a method for insertion into a sorted list. Both these methods are from Glib which is a low-level C library that forms basis of the GTK+ toolkit and the GNOME environment.\nThe methods \\emph{devres}\\footnote{method \\texttt{pcim\\_iounmap} in Linux kernel at \\texttt{linux\/lib\/devres.c}} and \\emph{rm\\_pkey}\\footnote{from InfiniBand device driver at \\texttt{drivers\/infiniband\/hw\/ipath\/ipath\\_mad.c}} are methods from the Linux kernel and an Infiniband device driver, both mentioned in~\\cite{wang-aplas10}.\n\n\n\n\nAll experiments were completed on an Intel Core i5 CPU at 2.4GHz with 6GB of RAM.\nFor all examples, our prototype implementation learns an adequate invariant really fast. Though the learned QDA might not be the smallest automaton representing the samples $S$ (because of the inaccuracies of the teacher),\nin practice we find that they are reasonably small (less than 50 states).\nMoreover, we verified that the learned invariants were adequate for proving\nthe programs correct by generating verification conditions and validating them using an SMT solver\n(these verified in less than 1s).\nLearnt invariants are complex in some programs; for example  the invariant QDA for the program \\emph{list-find} is presented in Appendix~E and corresponds to:\\\\\n{\\small\n$head \\neq nil \\wedge (\\forall y_1 y_2. head \\rightarrow^* y_1 \\rightarrow^* y_2 \\Rightarrow d(y_1) \\leq d(y_2)) \\wedge ((cur = nil \\wedge \\forall y_1. head \\rightarrow^* y_1 \\Rightarrow d(y_1) < k) \\vee (head \\rightarrow^* cur \\wedge \\forall y_1. head \\rightarrow^* y_1 \\rightarrow^+ cur \\Rightarrow d(y_1) < k))$.\n}\n\n\n\n\\subsection*{Formula Words}\n\nA \\emph{formula word} over $PV$, ${\\cal F}$, and $Y$ is a word over\n$(\\Pi^* \\times \\mathcal{F})$ where, as before, $\\Pi = \\Sigma \\times\n(Y \\cup \\{\\yblank\\})$ and each $p \\in PV$ and\n$y \\in Y$ occurs exactly once in the word. Note that a formula word\ndoes not contain elements of the data domain -- it simply consists of\nthe symbolic word that depicts the pointers into the list (modeled\nusing $\\Sigma$) and a valuation for the quantified variables in $Y$\n(modeled using the second component) as well as a formula over the\nlattice $\\mathcal{F}$. For example,\n$\\bigl( (\\{h\\},y_1) (b, \\yblank) (b, y_2) (\\{t\\}, \\yblank), d(y_1)\\leq d(y_2) \\bigr)$\nis a formula word, where $h$ points to the first element, $t$ to the last element, $y_1$ points to the first element, and $y_2$ to the third element; and the data formula is $d(y_1) \\leq d(y_2)$.\n\nBy using formula words we explicitly take the view of a QDA as a Moore machine that reads symbolic words and outputs data formulas. A formula word $(u, \\alpha)$ is accepted by a QDA $\\mathcal A$ if $\\mathcal A$ reaches the state $q$ after reading $u$ and $f(q) = \\alpha$. Hence, a QDA defines a unique language of formula words in the usual way.\nBy means of formula words, we can reduce the problem of learning QDAs to the problem of learning Moore machines.\nNext, we briefly sketch the learning framework we use for learning QDAs.\\\\\n\\noindent {\\bf Actively learning QDAs}:\nAngluin \\cite{DBLP:journals\/iandc\/Angluin87} introduced a popular learning framework in which a \\emph{learner}\nlearns a regular language $L$, the so-called \\emph{target language}, over an a priory fixed alphabet $\\Sigma$ by actively querying a \\emph{teacher} which is capable of answering  \\emph{membership} and \\emph{equivalence queries}.\nAngluin's algorithm\nlearns a regular language in time polynomial in the size of the (unique) minimal deterministic finite automaton accepting the target language and the length of the longest counterexample returned by the teacher.\nThis algorithm can however be easily lifted to the learning of Moore machines (see Appendix~B for details).\nMembership queries now ask for the output or classification of a word.\nOn an equivalence query, the teacher says ``yes'' or returns a counter-example $w$ such that the output of\nthe conjecture on $w$ is different from the output on $w$ in the target language.\nAs  QDAs can viewed as Moore languages (since it's just a set of words with output being data-formulas), \nwe can apply Angluin's algorithm directly in order to learn a QDA, and obtain the following theorem.\n\n\\vspace{-0.07cm}\n\\begin{theorem}\nGiven a teacher for a QDA-acceptable language of formula words that can answer membership and equivalence queries, the unique minimal QDA for this language can be learned in time polynomial in this minimal QDA and the length of the longest counterexample returned by the teacher.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection*{List and Array Invariants:}\n\nConsider a typical invariant in a sorting program over lists where the loop invariant is expressed as:\\\\\n$\\textit{head} \\rightarrow^* i ~~\\wedge ~~\\forall y_1, y_2. ((\\textit{head} \\rightarrow^* y_1 \\wedge \\textit{succ}(y_1, y_2) \\wedge y_2  \\rightarrow^* i ) \\Rightarrow d(y_1) \\leq d(y_2))~~~(1)$\\\\\nThis says that for all cells $y_1$ that occur somewhere in the list pointed to by $\\textit{head}$ and where $y_2$ is the successor of $y_1$, and where $y_1$ and $y_2$ are before the cell pointed to by a scalar pointer variable $i$, the data value stored at $y_1$ is no larger than the data value stored at $y_2$. \nThis formula is \\emph{not} in the decidable fragment of \\Strand since the universally quantified variables are involved in a non-elastic relation $\\textit{succ}$ (in the subformula $\\textit{succ}(y_1, y_2)$). Such an invariant for a program manipulating arrays can be expressed as:\\\\\n$~~~~~~~~~~~\\forall y_1, y_2. ((0 \\leq y_1 \\wedge y_2=y_1+1 \\wedge y_2 \\leq i) \\Rightarrow A[y_1] \\leq A[y_2])~~~~~~~~~~~~~~~~(2)$\\\\\nNote that the above formula is also not in the decidable array property fragment.\\\\\n\n{\\bf Quantified Data Automata:}\nThe key idea in this paper is an automaton model for expressing such constraints called \\emph{quantified data automata} (QDA). The above two invariants are expressed by the following QDA:\n\n\n\\begin{center}\n\t\\vspace{-3mm}\n\t\\begin{tikzpicture}\n\t\n\t\t\\node[state]                        (0) {$q_0$};\n\t\t\\node[state, right=2cm of 0]            (1) {$q_1$};\n\t\t\\node[state, right=2.5cm of 1]            (2) {$q_2$};\n\t\t\\node[state, right of=2]            (3) {$q_3$};\n\t\t\\node[state, right of=3, accepting] (4) {$q_4$};\n\t\t\\node[state, above right= 1cm and 0.5cm of 1, accepting] (5) {$q_5$};\n\n\t\n\t\t\\path (4) node[right, xshift=10] {$\\scriptstyle d(y_1) \\leq d(y_2)$}\n\t\t      -- (5) node[right,xshift=10,yshift=5]{$\\scriptstyle true$}\n\t\t;\n\t\n\t\t\\draw[<-, shorten <=1pt] (0.west) -- +(-.3, 0);\n\t\t\\draw[->] (0) edge                node       {$\\scriptstyle (\\mathrm{head}, \\yblank)$} (1)\n\t\t\t (0) edge[bend left] node[yshift=-3] {$\\scriptstyle (\\mathrm{\\{head, i\\}, *}), (\\mathrm{head}, y_2)$} (5)\n\t\t              edge[bend right=17] node[swap] {$\\scriptstyle (\\mathrm{head}, y_1)$}     (2)\n\t\t\t\t\t\t\t(1) edge                node       {$\\scriptstyle (b, y_1)$}                 (2)\n\t\t\t\t\t\t\t(1) edge node[right] {$\\scriptstyle (i, *), (b, y_2)$} \t\t    (5)\n\t\t\t\t\t\t\t\t\tedge[loop above]    node       {$\\scriptstyle \\blank$}                   ()\n\t\t\t\t\t\t\t(2) edge                node       {$\\scriptstyle (b, y_2)$}                 (3)\n\t\t\t\t\t\t\t(2) edge[bend right] node[right,xshift=2] {$\\scriptstyle \\blank,~ (i, \\yblank)$}  (5)\n\t\t\t\t\t\t\t    edge[bend right=25] node[swap] {$\\scriptstyle (i, y_2)$}                 (4)\n\t\t\t\t\t\t\t(3) edge                node       {$\\scriptstyle (i, \\yblank)$}             (4)\n\t\t\t\t\t\t\t\t\tedge[loop above]    node       {$\\scriptstyle \\blank$}                   ()\n\t\t\t\t\t\t\t(4) edge[loop above]    node       {$\\scriptstyle \\blank$}                   ()\n\t\t\t\t\t\t\t(5) edge[loop above]    node       {$\\scriptstyle *$}                   ()\n\t\t\t\t\t\t\t;\n\t\\end{tikzpicture}\n\\vspace{-3mm}\n\n\\end{center}\n\nThe above automaton reads (deterministically) data-words whose labels denote the positions\npointed to by the scalar pointer variables $\\textit{head}$ and\n$\\textit{i}$, as well as valuations of the quantified variables $y_1$\nand $y_2$. We use two \\emph{blank} symbols that indicate that no\nscalar variable (``$b$'') or no variable from $Y$ (``$\\yblank$'') is read\nin the corresponding component; moreover, $\\blank = (b, -)$.\nMissing transitions go to a sink state labeled \\textit{false}.\nThe above automaton accepts a data-word $w$ with a valuation $v$ for \nthe universally quantified variables $y_1$ and $y_2$  as follows: it stores the value\nof the data at $y_1$ and $y_2$ in two registers, and then checks\nwhether the formula annotating the final state it reaches holds for these data\nvalues. The automaton accepts the data word $w$ if for \\emph{all}\npossible valuations of $y_1$ and $y_2$, the automaton accepts the corresponding word with valuation.\nThe above automaton hence accepts precisely those set of\ndata words that satisfy the invariant formula.\\\\\n\n\\noindent{\\bf Decidable Fragments and Elastic Quantified Data Automata:}\nThe emptiness problem for QDAs is undecidable; in other words, the logical formulas that QDAs express fall into \nundecidable theories of lists and arrays. A common restriction in the array property fragment\nas well as the syntactic decidable fragments of \\Strand is that quantification is not permitted to be over \nelements that are only a \\emph{bounded} distance away. \nThe restriction allows quantified variables to only be related through \\emph{elastic}\nrelations (following the terminology of \\Strand~\\cite{strand,strandsas})\n\nFor instance, a formula equivalent to the formula in Eq.~1\nbut expressed in the decidable fragment\nof \\Strand over lists is:\\\\\n$\\textit{head} \\rightarrow^* i ~~\\wedge ~~\\forall y_1, y_2. ((\\textit{head} \\rightarrow^* y_1 \\wedge y_1 \\rightarrow^* y_2 \\wedge y_2  \\rightarrow^* i ) \\Rightarrow d(y_1) \\leq d(y_2))~~~(3)$\\\\\nThis formula compares data at $y_1$ and $y_2$ whenever $y_2$ occurs sometime after $y_1$, and this makes the formula fall in a decidable class.\nSimilarly, a formula equivalent to the formula Eq.~2 in the decidable array property fragment is:\\\\\n$~~~~~~~~~~~~~~~~~\\forall y_1, y_2. ((0 \\leq y_1 \\wedge y_1 \\leq y_2 \\wedge y_2 \\leq i) \\Rightarrow A[y_1] \\leq A[y_2])~~~~~~~~~~~~~~~~(4)$\\\\\nThe above two formulas are captured by a QDA that is the same as in the figure above, except that the $\\blank$-transition from $q_2$ to $q_5$ is replaced by a $\\blank$-loop  on $q_2$. \n\n\nWe identify a restricted form of quantified data automata, called \\emph{elastic quantified data automata} (EQDA) in Section~\\ref{sec:elastic}, which\nstructurally captures the constraint that quantified variables can be related only using elastic relations\n(like $\\rightarrow^*$ and $\\leq$).\nFurthermore, we show\nin Section~\\ref{sec:application} that EQDAs can be converted to formulas in the decidable fragment of \\Strand and the array property fragment, and hence expresses invariants that are amenable to decidable analysis across loop bodies.\n\nIt is important to note that QDAs are not necessarily a blown-up version of the\nformulas they correspond to. For a formula, the corresponding QDA can\nbe exponential, but for a QDA the corresponding formula can be\nexponential as well (QDAs are like BDDs, where there is sharing of common suffixes of constraints, which\nis absent in a formula).\n\n\n\n\n\\section{Introduction} \\label{sec:introduction}\n\\vspace{-0.2cm}\n\nSynthesizing invariants for programs is one of the most challenging problems in verification today.\nIn this paper, we are interested in using \\emph{learning} techniques to synthesize quantified\ndata-structure invariants.\n\nIn an \\emph{active} black-box learning framework, we look upon the invariant as a set of configurations of the program,\nand allow the learner to query the teacher for membership and equivalence queries on this set.\nFurthermore, we fix a particular representation class for these sets, and demand that the learner\nlearn the smallest (simplest) representation that describes the set. A learning algorithm that learns\nin time polynomial in the size of the simplest representation of the set is desirable.\nIn \\emph{passive} black-box learning, the learner is given a sample of examples and counter-examples\nof configurations, and is asked to synthesize the simplest representation that includes the examples\nand excludes the counter-examples. In general, several active learning algorithms that work in polynomial\ntime are known (e.g., learning regular languages represented as DFAs \\cite{DBLP:journals\/iandc\/Angluin87}) while passive polynomial-time\nlearning is rare (e.g., conjunctive Boolean formulas can be learned but general Boolean formulas cannot\nbe learned efficiently, automata cannot be learned passively efficiently)~\\cite{KearnsVazirani}.\n\nIn this paper, we build active learning algorithms for \\emph{quantified logical formulas describing\nsets of linear data-structures}. Our aim is to build algorithms that can learn formulas of the kind\n ``$\\forall y_1, \\ldots y_k ~\\varphi$'', where $\\varphi$ is quantifier-free, and that captures properties\n of arrays and lists (the variables range over indices for arrays, and locations for lists, and the\n formula can refer to the data stored at these positions and compare them using arithmetic, etc.).\nFurthermore, we show that we can build learning algorithms that learn  properties that are\nexpressible in known decidable logics. We then\nemploy the active learning algorithm in a \\emph{passive learning} setting where\nwe show that by building an imprecise teacher that answers the questions of the active learner,\nwe can build effective invariant generation algorithms that learn simply from a finite set of\nexamples.\n\n\\noindent{\\bf Active Learning of Quantified Properties using QDAs:}\n Our first technical contribution is a novel representation (normal form) for quantified properties of linear\n data-structures, called \\emph{quantified data automata} (QDA), and a polynomial-time active learning algorithm\n for QDAs.\n\n We model linear data-structures as \\emph{data words}, where each position is decorated with a finite alphabet modeling the program's pointer variables that point to that cell in the list or index variables that index into the cell of the array, and with data modeling the data value stored in the cell, e.g. integers.\n\nQuantified data automata (QDA) are a new model of automata over data words that are powerful enough to express \\emph{universally} quantified properties of data words. A QDA accepts a data word provided it accepts \\emph{all possible} annotations of the data word with valuations of a (fixed) set of variables $Y=\\{ y_1, \\ldots, y_k\\}$; for each such annotation, the QDA reads the data word, records the data stored at the positions pointed to by $Y$, and finally checks these data values against a data formula determined by the final state reached. QDAs are very powerful in expressing typical invariants of programs manipulating lists and arrays, including invariants of a wide variety of searching and sorting algorithms, maintenance of lists and arrays using insertions\/deletions, in-place manipulations that destructively update lists, etc.\n\nWe develop an efficient learning algorithm for QDAs. By using a combination of \\emph{abstraction} over a set of data formulas and Angluin's learning algorithm for DFAs~\\cite{DBLP:journals\/iandc\/Angluin87}, we build a learning algorithm for QDAs. We first show that for any set of valuation words (data words with valuations for the variables $Y$), there is a \\emph{canonical} QDA. Using this result, we show that learning valuation words can be reduced to learning \\emph{formula words} (words with no data but paired with data formulas), which in turn can be achieved using Angluin-style learning of Moore machines. The number of queries the learner poses and the time it takes is bound polynomial in the size of the canonical QDA that is learned. Intuitively, given a set of pointers into linear data structures, there are an exponential number of ways to permute the pointers into these and the universally quantified variables; the learning algorithm allows us to search this space using only polynomial time in terms of the\nactual permutations that figure in the set of data words learned.\n\n\\noindent {\\bf Elastic QDAs and a Unique Minimal Over-Approximation Theorem:}\nThe quantified properties that we learn in this paper (we can synthesize them from QDAs)\nare very powerful, and are, in general \\emph{undecidable}.\nConsequently, even if they are learnt in an invariant-learning application, we will be unable to \\emph{verify}\nautomatically whether the learnt properties are adequate invariants for the program at hand.\nThe goal of this paper is to also offer mechanisms to \\emph{learn invariants that are amenable to decision procedures}.\n\nThe second technical contribution of this paper is to identify a subclass of QDAs (called elastic QDAs) and show\ntwo main results for them: (a) elastic QDAs can be converted to \\emph{decidable} logical formulas, to the\narray property fragment when modeling arrays and the decidable \\Strand fragment when modeling lists;\n(b) a surprising \\emph{unique minimal over-approximation theorem} that says that for every QDA, accepting say a language $L$ of valuation-words,\nthere is a \\emph{minimal} (with respect to inclusion) language of valuation-words $L'$ that is accepted by an elastic QDA.\n\nThe latter result allows us to learn QDAs and then apply the unique minimal over-approximation (which is effective) to compute\nthe best over-approximation of it that can be expressed by elastic QDAs (which then is decidable). The result is proved by showing\nthat there is a unique way to minimally morph a QDA to one that satisfies the elasticity restrictions.\nFor the former, we identify a common property of the array property fragment and the syntactic decidable fragment of \\Strand, called \\emph{elasticity} (following the general terminology on the literature on \\Strand~\\cite{strand}). Intuitively, both the array property fragment and \\Strand prohibit quantified cells to be tested to be bounded distance away (the array property fragment does this by disallowing arithmetic expressions over the quantified index variables~\\cite{apf} and the decidable fragment of \\Strand disallows this by permitting only the use of $\\rightarrow^*$ or $\\rightarrow^+$ in order to compare quantified variables~\\cite{strand,strandsas}). We finally identify a \\emph{structural restriction} of QDAs that permits only elastic properties to be stated\nthat there is a unique way to minimally morph a QDA to one that satisfies the elasticity restrictions.\n\n\\noindent{\\bf Passive Learning of Quantified Properties:}\nThe active learning algorithm can itself be used in a verification framework, where the membership and equivalence queries are answered\nusing under-approximate and deductive techniques (for instance, for iteratively increasing values of $k$, a teacher can answer membership\nquestions based on bounded and reverse-bounded model-checking, and answer equivalence queries by checking if the invariant is adequate using\na constraint solver; see Appendix~D for details). In this paper, we do not pursue an implementation of active learning as above, but instead build a passive learning algorithm that uses the active learning algorithm.\n\nOur motivation for doing passive learning is that we believe (and we validate this belief using experiments)\nthat in many problems, a lighter-weight passive-learning algorithm which learns from a few randomly-chosen small data-structures is sufficient\nto divine the invariant. Note that passive learning algorithms, in general, often boil down to a guess-and-check algorithm of some kind, and often pay an exponential price in the property learned. Designing a passive learning algorithm using an active learning core allows us\nto build more interesting algorithms; in our algorithm, the inacurracies\/guessing is confined to the way the teacher answers\nthe learner's questions.\n\nThe passive learning algorithm works as follows. Assume that we have a finite set of configurations $S$, obtained from sampling\nthe program (by perhaps just running the program on various random small inputs). We are required to learn the simplest representation\nthat captures the set $S$ (in the form of a QDA). We now use an active learning algorithm for QDAs; membership questions are answered\nwith respect to the set $S$ (note that this is imprecise, as an invariant $I$ must include $S$ but need not be precisely $S$).\nWhen asked an equivalence query with a set $I$, we check whether $S \\subseteq I$; if yes, we can check if the invariant is adequate using a constraint-solver and the program.\n\nIt turns out that this is a good way to build a passive learning algorithm. First, enumerating random small data-structures\nthat get manifest at the header of a loop fixes for the most part the structure of the invariant, since the invariant is forced to be expressed as\na QDA. Second, our active learning algorithm for QDAs promises never to ask long membership queries (queried words are guaranteed to be less than the diameter of the  automaton), and often the teacher has the correct answers.\nFinally, note that the passive learning algorithm answers membership queries with respect to $S$; this is because we do not\nknow the true invariant, and hence err on the side of keeping the invariant semantically small.\nThis inaccuracy is common in most learning algorithms employed for verification (e.g, Boolean learning~\\cite{wang-aplas10}, compositional verification~\\cite{Giano,CAV05}, etc). This inaccuracy could lead to a non-optimal QDA being learnt, and is precisely\nwhy our algorithm need not work in time polynomial in the simplest representation of the concept (though it is polynomial in\nthe invariant  it finally learns).\n\nThe proof of the efficacy of the passive learning algorithm rests in the experimental evaluation.\nWe implement the passive learning algorithm (which in turn uses the active learning algorithm).  By building a teacher using dynamic test runs of the program and by pitting this teacher against the learner, we learn invariant QDAs, and then over-approximate them using EQDAs. These EQDAs are then transformed into formulas over decidable theories of arrays and lists. Using a wide variety  of programs manipulating arrays and lists, ranging from several examples\nin the literature involving sorting algorithms, partitioning, merging lists, reversing lists, and programs from the Glib list library,\nprograms from the Linux kernel, a device driver, and programs from a verified-for-security mobile application platform,\nwe show that we can effectively learn adequate quantified invariants in these settings.\nIn fact, since our technique is a black-box technique, we show that it can be used to infer pre-conditions\/post-conditions for methods as well.\n\n\\input{relatedwork}\n\n\n\\vspace*{-3.5mm}\n\\section{Overview} \\label{sec:overview}\n\\vspace{-0.2cm}\n\\input{overview}\n\\vspace{-3.5mm}\n\n\\section{Quantified Data Automata} \\label{sec:preliminaries}\n\n\n\\input{preliminaries}\n\n\n\n\n\\vspace{-0.2cm}\n\\section{Learning Quantified Data Automata} \\label{sec:learning}\n\\vspace{-0.2cm}\n{\\input{learning}}\n\n\\vspace{-0.55cm}\n\\section{Unique Over-approximation Using Elastic QDAs} \\label{sec:elastic}\n\\vspace{-0.2cm}\n\n\n\\input{elastification.tex}\n\n\\vspace{-0.4cm}\n\\section{Linear Data-structures to Words and EQDAs to Logics} \\label{sec:application}\n\\vspace{-0.15cm}\n\nIn this section, we sketch briefly how to model arrays and lists as data-words, and how to convert EQDAs to quantified\nlogical formulas in decidable logics.\n\n\n\\input{translation}\n\n\n\n\n\\vspace{-0.4cm}\n\\section{Implementation and Evaluation on Learning Invariants}\n\\label{sec:implementation}\n\\vspace{-0.15cm}\n\n\\input{implementation}\n{\\bf Future Work:}\n\\input{conclusion}\n\n\n\\vspace{-0.3cm}\n\\bibliographystyle{splncs}\n{\\footnotesize\n\n\\subsection*{Valuation words}\n\n\n\n\n\n\n\n\n\n\n\n\nWe are now ready to introduce the automaton model.  A quantified data\nautomaton (QDA) over a set of program variables $PV$, a data domain\n$D$, a set of universally quantified variables $Y$, and a formula\nlattice ${\\cal F}$ is of the form $\\A = (Q, q_0,\n\\Pi, \\delta, f)$ where \n$Q$ is a finite set of states, \n$q_0 \\in Q$ is the initial state, \n$\\Pi = \\Sigma \\times (Y \\cup \\{\\yblank\\})$, \n$\\delta: Q \\times \\Pi \\rightarrow Q$ is the transition function, and\n$f: Q \\rightarrow F$ is a \\emph{final-evaluation function} that\n   maps each state to a data formula.\nThe alphabet $\\Pi$ used in a QDA does not contain data. Words over\n$\\Pi$ are referred to as \\emph{symbolic words} because they do not contain\nconcrete data values. The symbol $(b,\\yblank)$ indicating that a\nposition does not contain any variable is denoted by $\\blank$.\n\nIntuitively, a QDA is a \\emph{register} automaton that reads the data\nword extended by a valuation that has a register for each $y \\in Y$,\nwhich stores the data stored at the positions evaluated for $Y$, and\nchecks whether the formula decorating the final state reached holds\nfor these registers. It accepts a data word $w \\in (\\Sigma \\times\nD)^*$ if it accepts \\emph{all possible} valuation words $v$ extending\n$w$ with a valuation over $Y$.\n\n We formalize this below.\nA configuration of a QDA is a pair of the form $(q,r)$ where $q \\in Q$\nand $r: Y \\rightarrow D$ is a partial variable assignment. The initial\nconfiguration is $(q_0, r_0)$ where the domain of $r_0$ is empty. For\nany configuration $(q,r)$, any letter $a \\in \\Sigma$, data value $d\n\\in D$, and variable $y \\in Y$ we define $ \\delta'((q,r), (a,d,y)) =\n(q',r')$ provided $\\delta(q,(a,y)) = q'$ and $r'(y') = r(y')$ for each\n$y' \\not= y$ and $r'(y) = d$, and\nwe let $ \\delta'((q,r),\n(a,d,\\yblank)) = (q',r)$ if $\\delta(q,(a,\\yblank)) = q'$.  We extend\nthis function $\\delta'$ to valuation words in the natural way.\n\nA valuation word $v$ is accepted by the QDA if $\\delta'((q_0,r_0), v)\n= (q,r)$ where $(q_0,r_0)$ is the initial configuration and $r \\models\nf(q)$, i.e., the data stored in the registers in the final\nconfiguration satisfy the formula annotating the final state reached.\nWe denote the set of valuation words accepted by $\\A$ as $\\Lval(\\A)$. We\nassume that a QDA verifies whether its input satisfies the constraints\non the number of occurrences of variables from $PV$ and $Y$, and that\nall inputs violating these constraints either do not admit a run\n(because of missing transitions) or are mapped to a state with final\nformula $\\false$.\n\nA data word $w$ is accepted by the QDA if for every valuation word\n$v$ such that the data word corresponding to $v$ is $w$, $v$ is\naccepted by the QDA. The language of the QDA, $L(\\A)$, is the set of\ndata words accepted by it.\n\n\n\n\n\n\\subsubsection*{Related Work:}\n{\\bf Related Work:}\nFor invariants expressing properties on the dynamic heap, \\emph{shape analysis} techniques are the most well known~\\cite{shapeanalysis}, where locations are classified\/merged using \\emph{unary} predicates (some dictated by the program and some given as instrumentation predicates by the user), and abstractions summarize all nodes with the same predicates into a single node. The data automata that we build also express an infinite set of linear data structures, but do so using automata, and further allow $n$-ary quantified relations between data elements.\nIn recent work,~\\cite{celia} describes an abstract domain, for analyzing list manipulating programs, that can capture quantified properties about the structure and the data stored in lists. This domain can be instantiated with any numerical domain for the data constraints and a set of user-provided patterns for capturing the structural constraints.\nHowever, providing these patterns for quantified invariants is in general a difficult task.\n\nIn recent years, techniques based on \\emph{Craig's interpolation}~\\cite{mcmillan03} have emerged as\na new method for invariant synthesis.\nInterpolation techniques,\nwhich are inherently white-box as well, are known for several theories, including\nlinear arithmetic, uninterpreted function theories, and even quantified properties\nover arrays and lists~\\cite{mcmillan06,mcmillan08,natasha,podelski}. These methods use different heuristics like\nterm abstraction~\\cite{natasha}, preferring smaller constants~\\cite{mcmillan06,mcmillan08} and use of existential ghost variables~\\cite{podelski}\nto ensure that the interpolant converges on an invariant from a \\emph{finite} set of spurious counter-examples.\nIC3~\\cite{ic3} is another white-box technique for generalizing inductive invariants from a set of counter-examples.\n\n\nA primary difference in our work, compared to all the work above, is that ours is a \\emph{black-box technique} that does not look at the code of the program, but synthesizes an invariant from a snapshot of examples and counter-examples\nthat characterize the invariant. The black-box approach to constructing invariants has both advantages and disadvantages. The main disadvantage is that information regarding what the program actually does is lost in invariant synthesis. However, this is the basis for its advantage as well---\nby \\emph{not} looking at the code, the learning algorithm promises to learn the sets with\nthe simplest representations in polynomial time, and can also be much more flexible.\nFor instance, even when the code of the program\nis complex, for example having non-linear arithmetic or complex heap manipulations that preclude\nlogical reasoning, black-box learning gives ways to learn simple invariants for them.\n\n\nThere are several black-box learning algorithms that have been explored in verification. Boolean formula learning has been investigated for finding quantifier-free program invariants~\\cite{bowyawwang}, and also extended to quantified invariants~\\cite{wang-aplas10}. However unlike us,~\\cite{wang-aplas10} learns a quantified formula given a set of data predicates as also the predicates which can appear in the guards of the quantified formula. \nRecently, machine learning techniques have also been explored~\\cite{NoriCAV}. \nVariants of the Houdini algorithm~\\cite{houdini} essentially use conjunctive Boolean\nlearning (which can be achieved in polynomial time) to learn conjunctive invariants over templates of atomic formulas (see also~\\cite{gulwani}).\nThe most mature work in this area is Daikon~\\cite{daikon}, which learns formulas over a template, by enumerating all formulas and checking \nwhich ones satisfy the samples, and where scalability is achieved in practice using several heuristics that reduce the enumeration space which is doubly-exponential. For quantified\ninvariants over data-structures, however, such heuristics aren't very effective, and Daikon often restricts learning only to formulas of very restricted\nsyntax, like formulas with a single atomic guard, etc. In our experiments Daikon was, for instance, not able to learn the loop invariant for the selection sort algorithm.\n\n\n\n\n\\subsection*{Modeling Lists and Arrays as Data Words}\n\\label{sec:modeling}\n\\vspace{-0.2cm}\n\nWe model a linear data structure as a word over ($\\Sigma \\times D$) with $\\Sigma = 2^{PV}$, where $PV$ is the set of pointer variables and $D$ is the data domain; scalar variables in the program are modeled as single element lists.\nThe encoding introduces a special pointer variable $nil$ which is always read in the beginning of the data word together with all other null-pointers in the configuration.\nFor arrays, the encoding also introduces  $nil\\_le\\_zero$ and $nil\\_geq\\_size$ which are read together with  all those index variables which are less than zero or which exceed the size of the respective array.\nThe data value at these variables is not important; they can be populated with any data value in $D$. Given a configuration, the corresponding data words read the scalar variables and the linear data structures one after the other, in some pre-determined order. In programs like copying one array to another, where both the arrays are read synchronously, the encoding models multiple data structures as a single structure over an extended data domain.\n\n\n\n\n\n\n\n\\vspace{-0.3cm}\n\\subsection*{From EQDAs to \\Strand and APF}\\label{sec:EQDA_to_logic}\n\\vspace{-0.2cm}\n\nNow we briefly sketch the translation from an EQDA $\\mathcal{A}$ to an equivalent formula $\\mathcal{T}(\\mathcal{A})$ in $\\Strand$ or the APF such that the set of data words accepted by $\\mathcal{A}$ corresponds to the program configurations $\\mathcal{C}$ which model $\\mathcal{T}(\\mathcal{A})$.\n\nGiven an EQDA $\\mathcal{A}$, the translation enumerates all simple paths in the automaton to an output state. For each such path $p$ from the initial state to an output state $q_p$, the translation records the relative positions of the pointer and universal variables\nas a structural constraint $\\phi_p$ and the formula $f_{\\mathcal{A}}(q_p)$ relating the data value at these positions. Each path thus leads to a universally quantified implication of the form $\\forall Y. ~\\phi_p \\Rightarrow f_{\\mathcal{A}}(q_p)$. All valuation words not accepted by the EQDA semantically go to the formula \\emph{false}, hence an additional conjunct $\\forall Y.~\\neg (\\bigvee_p \\phi_p) \\Rightarrow \\textit{false}$ is added to the formula. So the final formula ${ \\mathcal{T}(\\mathcal{A}) = \\bigwedge_p \\forall Y.~ \\phi_p \\Rightarrow f_{\\mathcal{A}}(q_p) ~\\bigwedge~ \\forall Y.~\\neg (\\bigvee_p \\phi_p) \\Rightarrow \\textit{false}}$.\nSee Appendix C for more details.\n\n\\begin{figure}\n\\vspace{-0.55cm}\n\t\\centering\n\t\\begin{tikzpicture}\n\t\n\t\t\\node[state]                        (0) {$q_0$};\n\t\t\\node[state, right of=0, xshift=23] (1) {$q_2$};\n\t\t\\node[state, right of=1]            (2) {$q_8$};\n\t\t\\node[state, right of=2]            (3) {$q_{18}$};\n\t\t\\node[state, right of=3, accepting] (4) {$q_{26}$};\n\t\n\t\t\\draw[<-] (0.west) -- +(-.3, 0);\n\t\t\\draw[->] (0) edge             node {\\scriptsize $(\\{ \\textit{cur}, \\textit{nil} \\}, \\yblank )$} (1)\n\t\t           (1) edge             node {\\scriptsize $(\\textit{h}, \\yblank)$}                          (2)\n\t\t           (2) edge             node {\\scriptsize $(\\textit{b}, y_1)$}                               (3)\n\t\t               edge[loop above] node {\\scriptsize $\\blank$}                                          ()\n\t\t           (3) edge             node {\\scriptsize $(\\textit{b}, y_2)$}                               (4)\n\t\t               edge[loop above] node {\\scriptsize $\\blank$}                                          ()\n\t\t           (4) edge[loop above] node {\\scriptsize $\\blank$}                                          ();\n\t\n\t\t\\node[right of=4, xshift=-22, inner sep=0] {\\scriptsize $\\varphi$};\n\t\t\\node[anchor=west, inner sep=0] at (-.1, -.8) {\\scriptsize $\\varphi \\coloneqq  \\begin{array}[t]{l} d(y_1) \\leq d(y_2) \\wedge d(y_1) < \\textit{k} \\wedge d(y_2) < \\textit{k} \\end{array}$};\n\t\\end{tikzpicture}\n\t\\caption{A path in the automata expressing the invariant of the program which finds a key $k$ in a sorted list.The full automaton is presented in Appendix ~E.\\label{fig:translation}}\n\\vspace{-0.3cm}\n\\end{figure}\n\n\\vspace{-0.3cm}\nWe next explain, through an example, the construction of the structural constraints $\\phi_p$. Consider program \\emph{list-find} which searches for a key in a sorted list. The EQDA corresponding to the loop invariant learned for this program is presented in Appendix E. One of the simple paths in the automaton (along with the associated self-loops on $\\blank$) is shown in Fig~\\ref{fig:translation}. The structural constraint $\\phi_p$ intuitively captures all valuation words which are accepted by the automaton along $p$; for the path in the figure $\\phi_p$ is $(cur =\nnil \\wedge h \\rightarrow^+ y_1 \\wedge y_1 \\rightarrow^+ y_2)$ and the formula $\\forall y_1 y_2. ~(cur =\nnil \\wedge h \\rightarrow^+ y_1 \\wedge y_1 \\rightarrow^+ y_2) \\Rightarrow (d(y_1) \\leq d(y_2) \\wedge d(y_1) < \\textit{k} \\wedge d(y_2) < \\textit{k})$ is the corresponding conjunct in the learned invariant. \n\nApplying this construction yields the following theorem.\n\n\\vspace{-0.05cm}\n\\begin{theorem} \\label{thm:correctness-STRAND-translation}\nLet $\\mathcal{A}$ be an EQDA, $w$ a data word, and $c$ the program configuration corresponding to $w$. If $w \\in \\mathcal{L}(\\mathcal{A})$, then $c \\models \\mathcal{T}(\\mathcal{A})$. Additionally, if $\\mathcal{T}(\\mathcal{A})$ is a $\\Strand$ formula, then the implication also holds in the opposite direction.\n\\end{theorem}\n\n\nAPF allows the universal variables to be related by $\\leq$ or $=$ and not $<$. Hence, along paths where $y_1 < y_2$, we over-approximate the structural constraint $\\phi_p$ to $y_1 \\leq y_2$ and, subsequently, the data formula $f_{\\mathcal{A}}(q_p)$ is abstracted to include $d(y_1) = d(y_2)$. This leads to an abstraction of the actual semantics of the QDA and is the reason Theorem~\\ref{thm:correctness-STRAND-translation} only holds in one direction for the APF. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStreaming algorithms for processing massive graphs have been studied for two decades \\cite{hrrav98}. In the \nmost traditional setting, the {\\em insertion-only model}, an algorithm receives a sequence of the edges of the \ninput graph in arbitrary order, and the objective is to solve a graph problem using as little space as possible. \nThe insertion-only model has received significant attention, and many problems, such as matchings (e.g. \\cite{kmm12,gkk12,kks14,kr16,kt17,k18,ps19,fhmrr20}), \nindependent sets (e.g. \\cite{hssw12,hhls16,cdk18,cdk19}), and subgraph counting (e.g. \\cite{kmss12,cj17,bc17}), \nhave since been studied in this model. See \\cite{m14} for an excellent survey.\n\nIn 2012, Ahn et al. \\cite{agm12} introduced the first techniques for addressing {\\em insertion-deletion} graph streams,\nwhere the input stream consists of a sequence of edge insertions and deletions. They showed that\nmany problems, such as \\textsf{Connectivity} and \\textsf{Bipartiteness}, can be solved using the same amount of space \nas in insertion-only streams up to poly-logarithmic factors. Various other works subsequently gave results of a similar \nflavor and presented insertion-deletion streaming algorithms with similar space complexity as their insertion-only\ncounterparts for problems including \\textsf{Spectral Sparsification} \\cite{kmmmnst20} and \\textsf{$\\Delta+1$-coloring} \n\\cite{ack19}. Konrad \\cite{k15} and \nAssadi et al. \\cite{akly16} were the first to give a separation result between the insertion-only graph stream model and the \ninsertion-deletion graph stream model: While it is known that a $2$-approximation to \\textsf{Maximum Matching} can be computed using space \n$\\mathrm{O}(n \\log n)$ in insertion-only streams, Konrad showed that space $\\Omega(n^{\\frac{3}{2} - 4 \\epsilon})$ is required \nfor an $n^{\\epsilon}$-approximation in insertion-deletion streams, and Assadi et al. gave a lower\nbound of $n^{2-3\\epsilon-o(1)}$ for such an approximation. Assadi et al. also presented an $\\tilde{\\mathrm{O}}(n^{2-3\\epsilon})$ space \nalgorithm that matches their lower bound up to lower order terms, which establishes that their lower bound is optimal \n(a different algorithm that matches this lower bound is given by Chitnis et al. \\cite{ccehmmv16}). \n\n\nBoth Konrad and Assadi et al. exploit an elegant connection between insertion-deletion streaming algorithms\nand linear sketches.\nAi et al. \\cite{ahlw16}, building on the work of Yi et al. \\cite{lnw14}, showed that insertion-deletion graph \nstreaming algorithms can be characterized as algorithms that essentially solely rely on the computation of linear sketches of the input \nstream. A consequence of this result is that space lower bounds for insertion-deletion streaming algorithms\ncan also be proved in the {\\em simultaneous model of communication}, since linear sketches can be implemented in this \nmodel. This provides an alternative to the more common approach of proving streaming lower bounds in the one-way model of \ncommunication. In particular, the lower bounds by Konrad and Assadi et al. are proved in the simultaneous model of communication.\n\nFrom a technical perspective, this model has various attractive features, however, it comes with a major disadvantage:\nThe characterization of Ai et al. only holds for insertion-deletion streaming algorithms that (1) are able to process \n``very long'' input streams, i.e., input streams of triple exponential length in $n$, the number of vertices of the \ninput graph, and (2) are able to process multi-graphs. In particular, this characterization does not hold for \ninsertion-deletion streaming algorithms that rely on the assumption that input streams are short and the graph \ndescribed by the input stream is always simple. Consequently, the lower bounds of Konrad and Assadi et al. \ndo not hold for such algorithms.\n\n\\vspace{0.1cm}\n\\textbf{Our Results.}\nIn this work, we prove an optimal space lower bound for \\textsf{Maximum Matching} in insertion-deletion streams \nvia the one-way two-party model of communication. Our lower bound construction yields insertion-deletion streams of \nlength $\\mathrm{O}(n^2)$ and does not involve multi-edges. Our lower bound therefore also holds for streaming algorithms that are \ndesigned for short input streams and simple graphs for which the characterization by Ai et al. does not hold.\nFurthermore, the optimal lower bound by Assadi et al. \\cite{akly16} only holds for streaming algorithms that\nnever output non-existing edges when the (randomized) algorithms fail. We do not require this restriction.\n\nOur lower bound method is simple and more widely applicable. Using the same method, we also give an optimal \nlower bound for \\textsf{Minimum Vertex Cover}, showing that computing a $n^\\epsilon$-approximation requires $\\Omega(n^{2-2\\epsilon})$\nspace. Assadi and Khanna mention in \\cite{ak17} that the $n^{2 - 3\\epsilon - o(1)}$ space lower bound \nfor \\textsf{Maximum Matching} given in \\cite{akly16} also applies to \\textsf{Minimum Vertex Cover}. Our lower bound\ntherefore improves on this result by a factor of $n^{{\\epsilon} + o(1)}$. Furthermore, we show that our lower bound is \noptimal up to a factor of $\\log n$: We give a very simple deterministic \ninsertion-deletion streaming algorithm for \\textsf{Minimum Vertex Cover} that uses space $\\mathrm{O}(n^{2-2\\epsilon} \\log n)$.\n\n\n\nWhile the main application of our lower bounds in the one-way two-party communication model are lower bounds for insertion-deletion \ngraph streaming algorithms, we believe that our lower bounds are of independent interest. Indeed, the one-way two-party communication \ncomplexity of \\textsf{Maximum Matching} without deletions has been addressed in \\cite{gkk12}, and our result can therefore also be understood\nas a generalization of their model to incorporate deletions.\n\n\n\\vspace{0.1cm}\n\\textbf{The Simultaneous Model of Communication.}\nThe lower bounds by Konrad \\cite{k15} and Assadi et al. \\cite{akly16} are proved in the simultaneous model of communication. \nIn this model,\na typically large number of parties $k$ hold not necessarily disjoint subsets of the edges of the input graph.\nEach party $P_i$ sends a message $M_i$ to a referee, who then outputs the result of the protocol. \nThe connection between insertion-deletion streaming algorithms and linear sketches by Ai et al. \\cite{ahlw16} then implies that \na lower bound on the size of any message $M_i$ yields a lower bound on the space requirements of any \ninsertion-deletion streaming algorithm. \n\nIn the lower bound of Assadi et al. \\cite{akly16} for \\textsf{Maximum Matching}, each party $P_i$ holds the edges $E_i$\nof a dense subgraph, which itself constitutes a {\\em Ruzsa-Szemer\\'{e}di graph}, i.e., a graph whose edge set can be partitioned \ninto large disjoint induced matchings. All previous streaming lower bounds for approximate \\textsf{Maximum Matching} rely on \nrealizations of Ruzsa-Szemer\\'{e}di graphs \\cite{gkk12,k15,akly16}. \nTheir construction is so that \nonly a single induced matching of every party $P_i$ is useful for the construction of a global large matching. \nDue to symmetry of the construction, the parties are unable to identify the important induced matching and therefore \nneed to send large messages that contain information about most of the induced matchings to the referee for them to \nbe able to compute a large global matching. Interestingly, none of the parties hold edge deletions in their construction. \n\n\\vspace{0.1cm}\n\\textbf{The One-way Model of Communication.}\nIn this paper, we give a lower bound in the one-way two-party model of communication. In this model, Alice holds\na set of edges $E$ of the input graph and sends a message $M$ to Bob. Bob holds a set of edge deletions $D \\subseteq E$\nand outputs a large matching in the graph spanned by the edges $E \\setminus D$. A standard reduction shows that \na lower bound on the size of message $M$ also constitutes a lower bound on the space requirements of an \ninsertion-deletion streaming algorithm.\nThe two models are illustrated in Figure~\\ref{fig:models}.\n\n\n\\begin{figure}[h!]\n\\begin{center}\n \\begin{tikzpicture}\n  \\node (ref) at (0,0) {\\textbf{Referee}};\n  \\node (out1) at (2,0) {result};\n  \\node (p1) at (-1.5, -1.5) {$\\mathbf{P_1}$};\n  \\node (p2) at (-0.5, -1.5) {$\\mathbf{P_2}$};\n  \\node (pdots) at (0.5, -1.5) {\\dots};\n  \\node (p3) at (1.5, -1.5) {$\\mathbf{P_k}$};\n  \n  \\node(e1) at (-1.9, -2) {$E_1 \\subseteq E$};\n  \\node(e2) at (-0.3, -2) {$E_2 \\subseteq E$};\n  \\node(e3) at (1.5, -2) {$E_k \\subseteq E$};\n  \n  \\draw[->] (ref) -- (out1) {};\n  \\draw[->] (p1) -- (ref) node [midway, left,xshift=-2pt,yshift=-1pt] {$M_1$};\n  \\draw[->] (p2) -- (ref) node [midway, right,yshift=-1pt,xshift=-1pt] {$M_2$};\n  \\draw[->] (p3) -- (ref) node [midway, right,yshift=-1pt, xshift=4pt] {$M_k$};\n  \n  \n  \\node (alice) at (5, -0.5) {\\textbf{Alice}};\n  \\node (bob) at (8, -0.5) {\\textbf{Bob}};\n  \\node (out2) at (9.7,-0.5) {result};\n  \n \n  \\draw[->] (alice) -- (bob) node [midway, above] {$M$};\n  \\draw[->] (bob) -- (out2) {};\n  \n  \\node (ae1) at (5, -1) {$E$};\n  \\node (be2) at (8, -1) {$D \\subseteq E$};\n \\end{tikzpicture}\n \\caption{The simultaneous (left) and the one-way two-party (right) models of communication. \\label{fig:models}}\n \\end{center}\n\\end{figure}\n\n\\vspace{0.1cm}\n\\textbf{Our Techniques.}\nTo prove our lower bound, we identify that an insertion-deletion streaming algorithm for \\textsf{Maximum Matching}\nor \\textsf{Minimum Vertex Cover} can be used to obtain a one-way two-party communication protocol for a\ntwo-dimensional variant of the well-known \\textsf{Augmented Index} problem that we denote by \\textsf{Augmented Bi-Index}, \nor \\textsf{BInd} in short.\nIn an instance of \\textsf{BInd}, Alice holds an $n$-by-$n$ binary matrix $A \\in \\{0, 1\\}^{n \\times n}$. \nBob is given a position $(x,y) \\in [n-k]^2$\nand needs to output the bit $A_{x,y}$. Besides $(x, y)$, he also knows the $k$-by-$k$ submatrix of $A$ with upper left\ncorner at position $(x, y)$, however with the bit at position $(x,y)$ missing - we will denote this $k$-by-$k$ submatrix \nwith $(x, y)$ missing by $A_{S(x, y)}$. We show that this problem has a one-way communication complexity of $\\Omega((n-k)^2)$\nby giving a reduction from the \\textsf{Augmented Index} problem.\n\nTo obtain a lower bound for \\textsf{Maximum Matching}, we show that Alice and Bob can construct a protocol for \\textsf{BInd}\ngiven an insertion-deletion streaming algorithm for \\textsf{Maximum Matching}. In our reduction, we will consider instances\nwith $k = n - \\Theta(n^{1-\\epsilon})$, for some $\\epsilon > 0$. Consider the following attempt: Suppose that the input matrix \n$A$ is a uniform random binary matrix and that $A_{x,y} = 1$ (we will get rid of these assumptions later). \nAlice and Bob interpret the matrix $A$ as the incidence matrix of a bipartite graph $G$. Bob interprets \nthe ``1'' entries in the submatrix $A_{S(x,y)}$ outside the diagonal, i.e., all ``1'' entries except those in positions\n$\\{ (x+j, y+j) \\ : \\ 0 \\le j < k \\}$, as edge deletions $F$. \nThe graph $G - F$ has a large matching: Since the diagonal of $A_{S(x,y)}$ is not deleted, and each entry in the diagonal is $1$\nwith probability $1\/2$, we expect that half of all potential edges in the diagonal of $S(x,y)$ are contained in $G - F$ and thus form a matching of size \n$\\Theta(k) = \\Theta(n - n^{1-\\epsilon})$. An $n^\\epsilon$-approximation algorithm for \n\\textsf{Maximum Matching} would therefore report $\\Omega(n^{1-\\epsilon})$ of these edges. Suppose that the\nalgorithm reported $\\Omega(n^{1-\\epsilon})$ uniform random edges from the diagonal in $A_{S(x, y)}$ (we will also get rid of this\nassumption). Then, by repeating this scheme $\\Theta(n^\\epsilon)$ times in parallel, with large constant probability\nthe edge corresponding to $A_{x,y}$ is reported at least once, which allows us to solve \\textsf{BInd}. \nThis reduction yields an optimal $\\Omega(n^{2-3\\epsilon})$ space lower bound for insertion-deletion streaming algorithm for \n\\textsf{Maximum Matching}, since $\\Theta(n^{\\epsilon})$ parallel executions are used to solve a problem that has a lower bound \nof $\\Omega((n-k)^2) = \\Omega(n^{2-2\\epsilon})$.\n\nIn the description above, we assumed that (1) $A$ is a uniform random binary matrix; (2) $A_{x,y} = 1$; and (3) the algorithm outputs\nuniform random positions from the diagonal of $A_{S(x, y)}$. To eliminate (1) and (2), Alice and Bob first sample\na uniform random binary matrix $X \\in \\{0, 1\\}^{n \\times n}$ from public randomness and consider the matrix obtained \nby computing the entry-wise XOR between $A$ and $X$, i.e., matrix $A \\oplus X$, instead. Observe that $A \\oplus X$ \nis a uniform random binary matrix (independently of $A$), and with probability $\\frac{1}{2}$, property (2), i.e., $(A \\oplus X)_{x,y} = 1$, holds. \nRegarding assumption (3), besides computing the XOR $A \\oplus X$, Alice and Bob also sample two random \npermutations $\\sigma_1, \\sigma_2: [n] \\rightarrow [n]$ from public randomness. Alice and Bob permute the rows \nand columns of $A \\oplus X$ using $\\sigma_1$ and $\\sigma_2$, respectively.\nThen, no matter which elements from the permuted relevant diagonal of $A \\oplus X$ are reported by the algorithm, \ndue to the random permutations, these elements could have originated from any other position in this diagonal. \nThis in turn makes every element along the diagonal equally likely to be reported, including the position $(x,y)$ \n(in the unpermuted) matrix that we are interested in.\n\nOur reduction for \\textsf{Minimum Vertex Cover} is similar but simpler. We show that only a constant number of parallel executions\nof an insertion-deletion streaming are required. \n\n\n\\vspace{0.1cm}\n\\textbf{Further Related Work.}\nHosseini et al. \\cite{hly19} were able to improve on the ``triple exponential length'' requirement of the input streams\nfor a characterization of insertion-deletion streaming algorithms in terms of linear\nsketches by Li et al. \\cite{lnw14} and Ai et al. \\cite{ahlw16}. They showed that in the case of XOR-streams \nand $0\/1$-output functions, input streams of length $\\mathrm{O}(n^2)$ are enough.\n\nVery recently, Kallaugher and Price \\cite{kp20} showed that if either the stream length or the maximum value \nof the stream (e.g. the maximum multiplicity of an edge in a graph stream) are substantially restricted, \nthen the characterization of turnstile streams as linear sketches cannot hold. For these situations they \ndiscuss problems where linear sketching is exponentially harder than turnstile streaming.\n\n\nBesides the \\textsf{Maximum Matching} problem, the only other separation result between the insertion-only and\nthe insertion-deletion graph stream models that we are aware of is a recent result by Konrad \\cite{k19}, who showed that \napproximating large stars is significantly harder in insertion-deletion streams.\n\n\n\\vspace{0.1cm}\n\\textbf{Outline.}\nWe give a lower bound on the communication complexity of \\textsf{Augmented Bi-Index} in Section~\\ref{sec:aug-bi-index}.\nThen, in Section~\\ref{sec:matching}, we show that a one-way two-party communication protocol for \\textsf{Maximum Matching} \ncan be used to solve \\textsf{Augmented Bi-Index}, which yield an optimal space lower bound for \\textsf{Maximum Matching} \nin insertion-deletion streams. We conclude with a similar reduction for \\textsf{Minimum Vertex Cover} in \nSection~\\ref{sec:vertex-cover}, which also implies an optimal space lower bound for \\textsf{Minimum Vertex Cover} in \ninsertion-deletion streams.\n\n\\section{Augmented Bi-Index} \\label{sec:aug-bi-index}\nIn this section, we define the one-way two-party communication problem \\textsf{Augmented Bi-Index} and prove a\nlower bound on its communication complexity. \n\n\\begin{problem}[\\textsf{Augmented Bi-Index}]\n    In an instance of {\\em \\textsf{Augmented Bi-Index}} {\\em $\\textsf{BInd}^{n,k}_\\delta$} we have two players denoted Alice and Bob.\n    Alice holds a binary matrix $A \\in \\lbrace 0,1 \\rbrace^{n \\times n}$. Bob holds indices $x, y \\in [n-k]$ and the incomplete\\footnote{We use $A_S$ to refer to the collection of entries indexed by the set $S$, so $A_S = (A_{i,j})_{(i,j)\\in S}$.} binary matrix $A_{S(x,y)}$ where\n    \\[\n        S(x,y) = \\{ (i,j) \\in [n]^2 \\,|\\, (x \\leq i < x + k) \\text{ and } (y \\leq j < y + k) \\} \\setminus \\{ (x,y) \\} \\ .\n    \\]    \n    Alice sends a single message $M$ to Bob who must output $A_{x, y}$ with probability at least $1-\\delta$.\n\\end{problem}\n\nOur lower bound proof consists of a reduction from the well-known \\textsf{Augmented Index} problem, which is known to \nhave large communication complexity.\n\n\n\n\n\\begin{problem}[\\textsf{Augmented Index}]\n    In an instance of {\\em \\textsf{Augmented Index}} {\\em $\\textsf{Ind}^{n}_\\delta$} we have two players denoted Alice and Bob.\n    Alice holds a binary vector $V \\in \\lbrace 0,1 \\rbrace^n$. Bob holds an index $\\ell \\in [n]$ and the vector suffix $V_{>\\ell} = (V_{\\ell+1}, V_{\\ell+2}, \\cdots, V_n)$. \n    Alice sends a single message $M$ to Bob who must output $V_\\ell$ with probability at least $1-\\delta$.\n\\end{problem}\n\nAs a consequence of Lemma~13 in \\cite{mnsw98}, we can see that this problem has linear communication complexity (see also Lemma~2 in \\cite{bjkk04} for a more direct proof technique).\n\n\\begin{theorem}[e.g. \\cite{mnsw98}]\\label{thm:augind-lower}\n    For $\\delta < 1\/3$, any randomised one-way communication protocol which solves $\\textsf{Ind}^{n}_\\delta$ must communicate $\\Omega(n)$ bits.\n\\end{theorem}\n\n\n\n\n\nWe are now ready to prove our lower bound for \\textsf{Augmented Bi-Index}.\n\\begin{theorem}\\label{thm:bind-lower}\n    For $\\delta < 1\/3$, any randomised one-way communication protocol which solves $\\textsf{BInd}^{n,k}_\\delta$ must communicate $\\Omega((n-k)^2)$ bits.\n\\end{theorem}\n\\begin{proof}\n    Let $\\mathcal{P}$ be a communication protocol for $\\textsf{BInd}^{n,k}_\\delta$ that uses messages of length at most \n    $S(n, k)$ bits. We will show how $\\mathcal{P}$ can be used to solve $\\textsf{Ind}^{(n-k)^2}_\\delta$ with the same message size.\n    \n    Let $V, \\ell$ be any instance of $\\textsf{Ind}^{(n-k)^2}_\\delta$. Alice builds the matrix $A \\in \\{ 0, 1 \\}^{n \\times n}$ by placing the bits of $V$ in lexicographical order in the top-left $(n-k)$-by-$(n-k)$ region:\n    \\[\n        A_{i,j} =   \\begin{cases}\n                        V_{j+(n-k)(i-1)} &\\text{for } i,j \\in [n-k]\\\\\n                        0 &\\text{otherwise}\n                    \\end{cases} \\ .\n    \\]    \n    This packing is illustrated in Figure~\\ref{fig:aug-index}(a).\n    \n    Alice runs protocol $\\mathcal{P}$ on $A$ and sends the resulting message $M$ to Bob.    \n    Now, Bob has the message $M$, the index $\\ell \\in [(n-k)^2]$ and the suffix $V_{>\\ell}$. Let $x, y \\in [n-k]$ be the unique pair of integers such that $\\ell = y + (n-k)(x-1)$. Observe that $A_{x,y} = V_\\ell$.\n    \n    For Bob to be able to complete protocol $\\mathcal{P}$ he needs to provide $A_{S(x,y)}$. Because of the way we packed the entries of $V$ onto $A$, the overlap between $V$ and $A_{S(x,y)}$ is a subset of the entries of $V_{>\\ell}$ (see Figure~\\ref{fig:aug-index}(b) for an illustration). Therefore Bob can complete the protocol and determine $A_{x,y} = V_\\ell$ with probability at least $1-\\delta$.    \n    By Theorem~\\ref{thm:augind-lower}, it must be that $S(n, k) = \\Omega((n-k)^2)$.\n\\end{proof}\n\n\\begin{figure}\n    \\centering\n    \\hspace{1cm}\n    \\subcaptionbox{Example packing of the bits of $V$ into matrix $A$ with $n=9$ and $k=4$.}[0.4\\textwidth]{\n        \\begin{tikzpicture}[\n            cell\/.style={rectangle, draw=black!35!white, minimum size =6mm, inner sep=0mm}\n        ]\n            \\fill[blue!20!white] (3mm, -3mm) rectangle (33mm, -33mm);\n            \\foreach \\y in {1,...,5}\n            {\n                \\foreach \\x in {1,...,5}\n                {                \n                    \\pgfmathtruncatemacro{\\i}{int(int((\\y-1)*5 + \\x))}\n                    \\node[cell] at (6*\\x mm, -6*\\y mm) {\\small $V_{\\i}$};\n                }\n                \\foreach \\x in {6,...,9} \\node[cell] at (6*\\x mm, -6*\\y mm) {$0$};\n            }\n            \\foreach \\y in {6,...,9}\n            {\n                \\foreach \\x in {1,...,9} \\node[cell] at (6*\\x mm, -6*\\y mm) {$0$};\n            }\n            \\draw[black, very thick] (3mm, -3mm) rectangle (57mm, -57mm);\n            \\draw[black, very thick] (3mm, -3mm) rectangle (33mm, -33mm);\n        \\end{tikzpicture}\n    }\n    \\hspace{1cm}\n    \\subcaptionbox{Bob can construct the area $A_{S(x,y)}$ given $V_{>\\ell}$, which is part of his input.}[0.4\\textwidth]{\n        \\begin{tikzpicture}[\n            cell\/.style={rectangle, minimum size =6mm, inner sep=0mm}\n        ]\n            \\fill[blue!20!white] (3mm, -3mm) rectangle (33mm, -33mm);\n            \\node[cell, fill=red!30!white] at (18mm, -18mm) {\\small $V_\\ell$};\n            \\fill[green!20!white] (3mm, -21mm) -- (3mm, -33mm) -- (33mm, -33mm) -- (33mm, -15mm) -- (21mm, -15mm) -- (21mm, -21mm) -- (3mm, -21mm);\n            \\draw[very thick, dashed] (15mm, -21mm) -- (15mm, -39mm) -- (39mm, -39mm) -- node[right] {$A_{S(x,y)}$} (39mm, -15mm) -- (21mm, -15mm) -- (21mm, -21mm) -- (15mm, -21mm);\n            \\draw[black, very thick] (3mm, -3mm) rectangle (57mm, -57mm);\n        \\end{tikzpicture}\n    }\n    \\caption{The construction of $A$ and $A_{S(x,y)}$ in Theorem~\\ref{thm:bind-lower}.}\n    \\label{fig:aug-index}\n\\end{figure}\n\n\\section{Maximum Matching} \\label{sec:matching}\nLet $\\mathbf{A}$ be a $C$-approximation insertion-deletion streaming algorithm for \\textsf{Maximum Matching} that errs \nwith probability at most $1\/10$. We will now show that $\\mathbf{A}$ can be used to solve $\\textsf{BInd}^{n,k}_{\\delta}$.\n\n\\subsection{Reduction}\nLet $A \\in \\{0, 1\\}^{n \\times n}, x \\in [n-k]$ and  $y \\in [n-k]$ be an instance of $\\textsf{BInd}^{n,k}_{\\delta}$. \nAlice and Bob first sample a uniform random binary matrix $X \\in \\{0,1\\}^{n \\times n}$ \nand random permutations $\\sigma_1, \\sigma_2: [n] \\rightarrow [n]$ from public randomness. Alice then computes matrix \n$A'$ which is obtained by first computing the entry-wise XOR of $A$ and $X$, denoted by $A \\oplus X$, \nand then by permuting the rows and columns\nof the resulting matrix by $\\sigma_1$ and $\\sigma_2$, respectively. Next, Alice interprets $A'$ as the incidence matrix\nof a bipartite graph $G(A')$. Alice runs algorithm $\\mathbf{A}$ on a random ordering of the edges of $G(A')$ and sends\nthe resulting memory state to Bob.\n\nNext, Bob also computes the entry-wise XOR between the part of the matrix $A$ that he knows about, $A_{S(x,y)}$, and $X$, followed by applying\nthe permutations $\\sigma_1$ and $\\sigma_2$. In doing so, Bob knows the matrix entries of $A'$ at positions \n$(\\sigma_1(i), \\sigma_2(j))$\nfor every $(i,j) \\in S(x,y)$. He can therefore compute the subset $E_S$ of the edges of $G(A')$ with\n\n$$E_S = \\{(\\sigma_1(i),\\sigma_2(j)) \\in [n]^2 \\ | \\ (i,j) \\in S(x,y) \\mbox{ and } A'(\\sigma_1(i),\\sigma_2(j)) = 1  \\} \\ .$$\n\nFurthermore, let $E_{diag} \\subseteq E_S$ be the set of edges $(\\sigma_1(i), \\sigma_2(j))$ so that $(i,j)$ lies on the same\ndiagonal in $A$ as $(x,y)$, or, in other words, there exists an integer $1 \\le q \\le k-1$ such that $(x+q, y+q) = (i, j)$.\nThen, let $E_{del} = E_S \\setminus E_{diag}$.\nBob continues the execution of algorithm $\\mathbf{A}$, as follows: for every edge $e \\in E_{del}$, \nBob introduces an edge deletion of $e$, in random order.\n\nLet $M'$ be the matching returned by $\\mathbf{A}$.\nFrom $M'$ Bob computes the matching $M$ as follows: If $|M' \\le 0.99 \\frac{k}{2C}|$ then Bob sets $M = \\varnothing$.\nOtherwise, Bob sets $M$ to be a uniform random subset of $M'$ of size exactly $0.99 \\frac{k}{2C}$.\n\n\\textbf{Parallel Executions.} Alice and Bob execute the previous process $\\ell = 100 \\cdot C$ times in parallel. Let $M^i$, $X^i$, \n$\\sigma_1^i$ and $\\sigma_2^i$ be $M$, $X$, $\\sigma_1$ and $\\sigma_2$ that are used in run $i$, respectively.\nLet $Q_i$ be the indicator random variable that is $1$ iff $M^i$ contains the edge $(\\sigma^i_1(x), \\sigma^i_2(y))$. \nWe also define $p = \\sum_i Q_i$ to be the total number of times the edges $(\\sigma^i_1(x), \\sigma^i_2(y))$ are reported. \nWhenever the edge $(\\sigma^i_1(x), \\sigma^i_2(y))$ is reported, we interpret this to be a claim that $A_{x,y} = \\neg X^i_{x,y}$. So depending on the value of $X^i_{x,y}$, this acts as a claim that $A_{x,y} = 0$ or $A_{x,y} = 1$. We define $p_0 = \\sum_{i: Q_i = 1} X^i_{x, y}$ (which counts how often\n$A_{x,y} = 0$ was claimed) and let $p_1 = p - p_0$ (the number of times $A_{x,y} = 1$ was claimed). Bob outputs $1$ as his estimator for $A_{x,y}$ if $p_1 \\ge p_0$ \nand $0$ otherwise.\n\n\n\\subsection{Analysis}\nLet $G$ be the bipartite graph with incidence matrix $A \\oplus X$, and let \n$$F = \\{ (i,j) \\in S(x,y) \\ | \\ (A \\oplus X)_{i,j} = 1 \\mbox{ and } \\nexists \\ q \\mbox{ s.t. } (i,j) = (x+q, y+q) \\} \\ .$$\nThen the graph $G - F$ is isomorphic to the graph $G(A') - E_{del}$. In particular, \n$G(A') - E_{del}$ is obtained from $G - F$ by relabeling the vertex sets of the two bipartitions\nusing the permutations $\\sigma_1$ and $\\sigma_2$. \n\nWe will first bound the maximum matching size in $G(A') - E_{del}$. To this end, we will bound the \nmaximum matching size in $G - F$, which is easier to do:\n\n\\begin{lemma}\\label{lem:matching-size}\n With probability $1-\\frac{1}{k^{10}}$, the graph $G(A') - E_{del}$ is such that:\n $$0.99 \\frac{k}{2} \\le \\mu(G(A') - E_{del}) \\le 1.01 \\frac{k}{2} + 2(n-k) \\ ,$$\n where $\\mu(G)$ denotes the \\emph{matching number} of $G$, i.e., the size of a maximum matching.\n\\end{lemma}\n\\begin{proof}\n We will consider the graph $G - F$ instead, since it is isomorphic to $G(A') - E_{del}$ and has the same\n maximum matching size.\n \n First, observe that $G$ is a random bipartite graph where every edge is included with probability $\\frac{1}{2}$.\n Let $U$ and $V$ denote the bipartitions in $G$, and consider the subsets $U' = [x, x+k)$ and $V' = [y, y+k)$.\n Observe that in the vertex induced subgraph $G[U' \\cup V']$\n all edges are deleted in $F$ except those that connect \n the vertices $x+i$ and $y+i$, for every $0 \\le i \\le k-1$. By a Chernoff bound, the number of edges and thus\n the maximum matching size in $G[U' \\cup V']$ is bounded by:\n $$0.99 \\cdot \\frac{k}{2} \\le \\mu(G[U' \\cup V']) \\le 1.01 \\cdot \\frac{k}{2} \\ , $$\n with probability $1 - \\frac{1}{k^{10}}$.\n \n Observe that, with probability $1 - \\frac{1}{k^{10}}$, the neighborhood $\\Gamma(U')$ is such that \n $$0.99 \\cdot \\frac{k}{2} \\le |\\Gamma(U')| \\le 1.01 \\cdot \\frac{k}{2} + (n-k) \\ .$$ \n The set $U'$ can therefore be matched to at most $1.01 \\cdot \\frac{k}{2} + (n-k)$ vertices in $V$. \n We thus obtain\n $$\\mu(G - F) \\le 1.01 \\cdot \\frac{k}{2} + 2(n-k) \\ ,$$\n since we may also be able to match all $n-k$ vertices of $U \\setminus U'$. \n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:learn}\n Suppose that $M_i \\neq \\varnothing$. Then:\n \\begin{eqnarray*}\n\\frac{0.99}{2C} - \\frac{2(n - k)}{k} & \\le & \\Pr \\left[ Q_i = 1 \\right] \\le \\frac{0.99}{2C} \\ .\n \\end{eqnarray*}\n \n\n\\end{lemma}\n\\begin{proof}\n First, by construction of our reduction, since $M_i \\neq \\varnothing$ we have \n $|M_i| = 0.99 \\frac{k}{2C}$. \n Let \n $$U'_i = \\sigma_1^i([x, x+k)) \\mbox{ and } V'_i = \\sigma_2^i([y, y+k)) \\ .$$ \n Let $\\tilde{M}_i$ be the set of edges of $M_i$ connecting vertices in $U'_i$ to $V'_i$. Observe that there are\n $2(n-k)$ vertices in the graph outside the set $U'_i \\cup V'_i$. We thus \n have  \n $$|M_i| - 2(n-k) \\le |\\tilde{M}_i| \\le |M_i|  \\ .$$ \n Next, since the permutations $\\sigma^i_1, \\sigma^i_2$ are chosen uniformly at random, any edge of $\\tilde{M}_i$\n may have originated from any of the diagonal entries in $A_{S(x, y)}$. Hence, $\\tilde{M}_i$ claims the bits\n of at least $|M_i| - 2(n-k)$ and at most $|M_i|$ uniform random positions in the diagonal of $A_{S(x, y)}$.\n Every entry in the diagonal of $A_{S(x,y)}$ is thus claimed with the same probability.\n Since the diagonal of $A_{S(x,y)}$ is of length $k$, this probability is at least\n $$ \n \\frac{|M_i| - 2(n-k)}{k} = \\frac{0.99 \\frac{k}{2C} - 2(n-k)}{k} = \\frac{0.99}{2C} - \\frac{2(n - k)}{k} \\ ,\n $$\n and at most\n $$\\frac{|M_i|}{k} = \\frac{0.99 \\frac{k}{2C}}{k} = \\frac{0.99}{2C} \\ .$$ \n\\end{proof}\n\n\n\n\n\n\n\n\\begin{theorem}\\label{thm:reduction} \n Let $\\mathbf{A}$ be a $n^{\\epsilon}$-approximation insertion-deletion streaming algorithm \n for \\textsf{Maximum Matching} that errs with probability at most $1\/10$ and uses space $s$. Then there exists a communication \n protocol for $\\textsf{BInd}^{n,n - \\frac{1}{40} n^{1-\\epsilon}}_{0.05}$ that communication $\\mathrm{O}(n^\\epsilon \\cdot s)$ bits.\n\\end{theorem}\n\\begin{proof}\nLet $C = n^{\\epsilon}$ and let $k = n - \\frac{1}{40} n^{1-\\epsilon}$.\n First, by Lemma~\\ref{lem:matching-size}, with probability $1-\\frac{1}{k^{10}}$, the graph \n $G(A') - E_{del}$ contains a matching of size at least $0.99 k\/ 2$. By a union bound, \n the probability that this graph is of at least this size in each of the $\\ell$ iterations  \n is at least $1 - \\frac{\\ell}{k^{10}}$. Suppose from now on that this event happens.\n\n Let $\\ell_1$ be the number of times the algorithm $\\mathbf{A}$ succeeds, and let $\\ell_0$ be the number of times $\\mathbf{A}$\n errs. Then, $\\ell = \\ell_0 + \\ell_1$. Whenever $\\mathbf{A}$ succeeds, since $\\mathbf{A}$ is a $C$-approximation\n algorithm, the matching $M'_i$ is of size $0.99 \\frac{k}{2C}$, which further implies that $M_i$ is of size exactly\n $0.99 \\frac{k}{2C}$. Since the algorithm must return a correct matching, every time we have a claim (i.e. $Q_i = 1$), the claimed bit value must be correct. Thus, by Lemma~\\ref{lem:learn}, we get a correct claim on $A_{x,y}$ with probability at least \n \\begin{eqnarray*}\n  \\frac{0.99}{2C} - \\frac{2(n - k)}{k} & = & \\frac{0.99}{2n^{\\epsilon}} - \\frac{2(\\frac{1}{40} n^{1-\\epsilon})}{n - \\frac{1}{40} n^{1-\\epsilon}} \\ge  \\frac{0.99}{2n^{\\epsilon}} - \\frac{\\frac{1}{40} n^{1-\\epsilon}}{n} = \\frac{0.99}{2n^{\\epsilon}} - \\frac{1}{40n^{\\epsilon}} \\ge \\frac{2}{5 n^\\epsilon}   \\ , \\label{eqn:018}\n \\end{eqnarray*}\nwhere we used the inequality $\\frac{2x}{y-x} \\ge \\frac{x}{y}$, which holds for every $y > x$. \nWe thus expect to see the correct bit claimed at least $\\ell_1 \\cdot \\frac{2}{5 n^\\epsilon}$ times in total.\nOn the other hand, incorrect claims of the bit value can only occur when the algorithm errs. In the worst case, $\\mathbf{A}$ \nwill make as many false claims as possible - so we assume the algorithm never results in $M_i = \\emptyset$ when it errs. \nLemma~\\ref{lem:learn} also allows us to bound the probability of an incorrect claim for this bad algorithm by $\\frac{0.99}{2n^\\epsilon}$. We thus expect to see the wrong bit value claimed at most $\\ell_0 \\cdot \\frac{0.99}{2C} \\le \\frac{\\ell_0}{2n^\\epsilon}$ times.\n \n Recall that $\\ell = 100 n^\\epsilon$. Then, by standard concentration bounds, the probability that \n $\\ell_0 \\ge 2 \\cdot \\frac{\\ell}{10}$ is at most $\\frac{1}{100}$ (recall that the error probability of $\\mathbf{A}$ \n is at most $\\frac{1}{10}$). Suppose now that $\\ell_0 \\le \\frac{1}{5} \\ell$ holds, which also implies that $\\ell_1 \\ge \\frac{4}{5} \\ell$.\n We thus expect to learn the correct bit at least \n $$\\frac{4}{5} 100 n^{\\epsilon} \\cdot \\frac{2}{5 n^\\epsilon} = 32$$\n times, and using a Chernoff bound, it can be seen that the probability that we learn the correct bit less than \n $21$ times is at most $0.02$. \n Similarly, we expect to learn the incorrect bit at most \n $$\\frac{1}{5} 100 n^{\\epsilon} \\cdot \\frac{1}{2n^\\epsilon} = 10$$\n times, and by a Chernoff bound, it can be seen that the probability that we learn the incorrect bit at least $20$ times \n is at most $ 0.01$. Our algorithm therefore succeeds if all these events happen. Taking a union bound\n over all failure probabilities that occurred in this proof, we see that our algorithm succeeds with probability\n $$1 - \\frac{100n^{\\epsilon}}{k^{10}} - 0.01 - 0.02 - 0.01 \\ge 0.95 \\ .$$\n\\end{proof}\n\nSince by Theorem~\\ref{thm:bind-lower}, $\\textsf{BInd}^{n,n - \\frac{1}{40} n^{1-\\epsilon}}_{0.05}$ has randomized one-way \ncommunication complexity $\\Omega(n^{2-2\\epsilon})$, by Theorem~\\ref{thm:reduction} we obtain our main result of this section\n\\begin{corollary}\n Every insertion-deletion $n^\\epsilon$-approximation streaming algorithm  \n for \\textsf{Maximum Matching} that errs with probability at most $\\frac{1}{10}$ requires space $\\Omega(n^{2-3\\epsilon})$.\n\\end{corollary}\n\n\n\n\n\\section{Minimum Vertex Cover} \\label{sec:vertex-cover}\nLet $\\mathbf{B}$ be a $C$-approximation insertion-deletion streaming algorithm for \\textsf{Minimum Vertex Cover} that succeeds with\nprobability $1 - 1\/400$. Similar to the previous section, we will now show how $\\mathbf{B}$ can be used to solve $\\textsf{BInd}^{n,k}_{\\delta}$.\n\n\\subsection{Reduction}\nLet $A \\in \\{0, 1\\}^{n \\times n}, x \\in [n-k]$ and  $y \\in [n-k]$ be an instance of $\\textsf{BInd}^{n,k}_{\\delta}$. \nThe reduction for \\textsf{Minimum Vertex Cover} is very similar to the reduction for \\textsf{Maximum Matching} presented \nin the previous section. Alice's behaviour is in fact identical:\n\nFirst, Alice and Bob sample a uniform random binary matrix $X \\in \\{0,1\\}^{n \\times n}$ and random permutations \n$\\sigma_1, \\sigma_2: [n] \\rightarrow [n]$ from public randomness. \nAlice then computes matrix $A'$ which is obtained by first computing $A \\oplus X$\nand then  permuting the rows and then the columns of the resulting matrix by $\\sigma_1$ and $\\sigma_2$, respectively. \nAlice interprets $A'$ as the incidence matrix of a bipartite graph $G(A')$. Alice then runs algorithm $\\mathbf{B}$ \non a random ordering of the edges of $G(A')$ and sends the resulting memory state to Bob.\n\nNext, Bob also computes the entry-wise XOR between the part of the matrix $A$ that he knows about and $X$, followed by applying\nthe permutations $\\sigma_1$ and $\\sigma_2$. In doing so, Bob knows the matrix entries of $A'$ at positions \n$(\\sigma_1(i), \\sigma_2(j))$\nfor every $(i,j) \\in S(x,y)$. He can therefore compute the subset $E_S$ of the edges of $G(A')$ with\n\n$$E_S = \\{(\\sigma_1(i),\\sigma_2(j)) \\in [n]^2 \\ | \\ (i,j) \\in S(x,y) \\mbox{ and } A'(\\sigma_1(i),\\sigma_2(j)) = 1  \\} \\ .$$\n\nNext, Bob continues the execution of $\\mathbf{B}$ and introduces deletions {\\em for all edges in $E_S$} in random order. \nObserve that this \nstep is different to the reduction for \\textsf{Maximum Matching}. Let $I$ be the vertex cover produced by $\\mathbf{B}$.\n\n\\textbf{Parallel Executions.} Alice and Bob run the procedure above $40$ times in parallel. Denote by $I^i$, $X^i$, $E^i_S$,\n$A'^i$, $\\sigma_1^i$, and $\\sigma_2^i$ the variables $I, X, E_S, A', \\sigma_1$ and $\\sigma_2$ used in iteration $i$.  \nFurthermore, let $Q_i$ be \nthe indicator variable that is $1$ iff $\\{ \\sigma^i_1(x), \\sigma^i_2(y) \\} \\cap I_i \\neq \\varnothing$,\ni.e., the potential edge $(\\sigma^i_1(x), \\sigma^i_2(y))$ is covered by the vertex cover. \n\nIf there exists a run $j$ with $Q_j = 0$, then Bob predicts $A_{x,y} = X_{x,y}$ (if there are multiple such runs\nthen Bob breaks ties arbitrarily). Otherwise, Bob returns \\texttt{fail} and the algorithm errs.\n\n\\subsection{Analysis}\n\nThe first lemma applies to every parallel run $j$. For simplicity of notation, we will omit\nthe superscripts that indicate the parallel run in our random variables.\n\nWe first show an upper bound on the size of a minimum vertex cover in $G(A') - E_S$.\n\n\\begin{lemma}\\label{lem:vc-size}\n The size of a minimum vertex cover in $G(A') - E_S$ is at most $2(n-k) + 1$.\n\\end{lemma}\n\\begin{proof}\nLet $U, V$ be the bipartitions of the graph $G(A') - E_S$, let $U' = \\{\\sigma_1(a) \\ : \\ a \\in [x, x+k) \\}$ and let\n$V' = \\{\\sigma_2(b) \\ : \\ b \\in [y, y+k) \\}$. Observe that $(G(A') - E_S)[U' \\cup V']$ \ncontains at most one edge: The potential edge between $\\sigma_1(x)$ and $\\sigma_2(y)$. A valid vertex cover \nof $G(A') - E_S$ is therefore $(U \\setminus U') \\cup (V \\setminus V') + \\sigma_1(x)$,\nwhich is of size $2(n-k) + 1$.\n\\end{proof}\n\nNext, we prove the key property of our reduction: We show that if $A'_{\\sigma_1(x), \\sigma_2(y)} = 0$ (or equivalently, \n$A_{x,y} \\oplus X_{x,y} = 0$)\nthen neither $\\sigma_1(x)$ nor $\\sigma_2(y)$ is in the output vertex cover with large probability.\n\n\\begin{lemma} \\label{lem:covered}\n Assume that algorithm $\\mathbf{B}$ does not err in run $j$. Suppose that \n $A'^j_{\\sigma^j_1(x), \\sigma^j_2(y)} = 0$. Then the probability that $Q_j = 1$ is at most \n $$\\frac{3 C \\cdot (2(n-k) + 1)}{k} \\ .$$\n\\end{lemma}\n\\begin{proof}\n Consider the set $D  = \\{ (\\sigma^j_1(x + i), \\sigma^j_2(y+i)) \\ | \\ 0 \\le i \\le k-1 \\}$, i.e., the positions of the diagonal\n of $S(x,y) \\cup \\{x, y\\}$ permuted by $\\sigma^j_1$ and $\\sigma^j_2$. Then, since $A'^j$ is a uniform random matrix, with probability\n at least $1-\\frac{1}{k^{10}}$, the ``permuted diagonal'' $A'^j_{D}$ contains at least $0.99 k\/ 2$ entries with value $0$,\n or, in other words, graph $G(A'^j) - E^j_S$ contains at least $0.99 k\/ 2$ non-edges in the positions of the permuted diagonal $D$.\n By Lemma~\\ref{lem:vc-size}, the size of a minimum vertex cover in $G(A'^{j}) - E^{j}_S$ is at most $2(n-k) + 1$, and since\n $\\mathbf{B}$ has an approximation factor of $C$, the  vertex cover $I_j$ is of size at most $C \\cdot (2(n-k) + 1)$. Hence, \n at most $C \\cdot (2(n-k) + 1)$ non-edges in $D$ can be covered in $I_j$. However, since the permutations are random, the \n probability that the non-edge $(\\sigma^j_1(x), \\sigma^j_2(y))$\n is covered, which is identical to the event $Q_j = 1$, is therefore at most \n $$\\frac{C \\cdot (2(n-k) + 1)}{0.99 k\/ 2} \\le \\frac{3 C \\cdot (2(n-k) + 1)}{k} \\ .$$\n\\end{proof}\n\n\n\\begin{theorem} \\label{thm:vc-reduction}\n Let $\\mathbf{B}$ be a $n^{\\epsilon}$-approximation \n insertion-deletion streaming algorithm for \\textsf{Minimum Vertex Cover} that uses space $s$ and errs with probability \n at most $1\/400$. Then, there exists a communication protocol for $\\textsf{BInd}^{n,n - \\frac{1}{20}n^{1-\\epsilon}}_{\\frac{1}{3}}$ \n that communicates $\\mathrm{O}(s)$ bits.\n\\end{theorem}\n\\begin{proof}\nLet $k = n - \\frac{1}{40} n^{1-\\epsilon}$ and let $C= n^{\\epsilon}$.\n Consider the reduction given in the previous subsection. First, observe that since $\\mathbf{B}$ errs with probability\n at most $1\/400$, by the union bound the probability that $\\mathbf{B}$ errs at least once in the $40$ parallel executions of our\n reduction is at most $\\frac{1}{10}$. We assume from now on that the algorithm never errs.\n \n Observe that the matrices $A'^j$ are random matrices. Hence,\n\n\n the probability that there exists at least one run $i$ with $A'^i_{\\sigma^i_1(x), \\sigma^i_2(y)} = 0$ is at least \n $1 - (\\frac{1}{2})^{40}$. Suppose that this event happens. Let run $i$ be so that $A'^i_{\\sigma^i_1(x), \\sigma^i_2(y)} = 0$.\n Then, by Lemma~\\ref{lem:covered}, the probability that the non-edge $(\\sigma^i_1(x), \\sigma^i_2(y))$ is covered by $I_i$,\n or in other words, the probability that $Q_i = 1$, is at most \n \n $$\\frac{3 C \\cdot (2(n-k) + 1)}{k} = \\frac{3n^{\\epsilon} \\cdot (\\frac{1}{20} n^{1-\\epsilon} + 1) }{n - \\frac{1}{40}n^{1-\\epsilon}} = \n \\frac{\\frac{3}{20}n + 3n^{\\epsilon}}{n - \\frac{1}{40}n^{1-\\epsilon}} = \\frac{3}{20} + o(1) \\ . $$  \n \n Observe that whenever $Q_i = 0$, the algorithm outputs $X^i_{x,y}$ as a predictor for $A_{x,y}$. Since the algorithm $\\mathbf{B}$ \n does not err, we have $A_{x,y} \\oplus X^i_{x,y} = 0$. This implies that $A_{x,y} = X^i_{x,y}$, which establishes correctness.\n \n Last, we need to bound the error probability of our algorithm. First, the probability that at least one of the $40$ runs \n fails is at most $\\frac{1}{10}$. Next, the probability that none of the runs are such that $A'^j_{\\sigma^j_1(x), \\sigma^j_2(y)} = 0$\n is at most $(\\frac{1}{2})^{40}$. Furthermore, the probability that $Q_i = 1$ when $A'^i_{\\sigma^i_1(x), \\sigma^i_2(y)} = 0$ is at \n most $\\frac{3}{20} + o(1)$. Applying the union bound, we see that the overall error probability of our algorithm is at most\n $$\\frac{1}{10}+ (\\frac{1}{2})^{40} + \\frac{3}{20} + o(1) \\le \\frac{1}{3} \\ , $$\n for large enough $n$. \n\\end{proof}\nSince by Theorem~\\ref{thm:bind-lower}, $\\textsf{BInd}^{n,n - \\frac{1}{40}n^{1-\\epsilon}}_{\\frac{1}{3}}$ has a \ncommunication complexity of $\\Omega(n^{2-2\\epsilon})$, we\nobtain the following result:\n\\begin{corollary}\n Every insertion-deletion $n^\\epsilon$-approximation streaming algorithm for \\textsf{Minimum Vertex Cover} with error probability \n at most $\\frac{1}{400}$ requires space $\\Omega(n^{2-2\\epsilon})$.\n\\end{corollary}\n\n\\subsection{Insertion-deletion Streaming Algorithm for \\textsf{Minimum Vertex Cover}}\nWe now sketch a simple deterministic $n^\\epsilon$-approximation insertion-deletion streaming algorithm for \n\\textsf{Minimum Vertex Cover} on general graphs that uses space $\\mathrm{O}(n^{2-2\\epsilon} \\log n)$. \nLet $G=(V, E)$ be the graph described by the input stream. The algorithm proceeds as follows: \n\\begin{enumerate}\n \\item Arbitrarily partition $V$ into subsets $V_1, V_2, \\dots, V_{n^{1-\\epsilon}}$,\n each of size $n^{\\epsilon}$.\n \\item Consider the multi-graph $G'$ obtained from $G$ by contracting the sets $V_i$ into vertices.\n \\item \\textbf{While processing the stream:} For each pair of vertices $V_i, V_j$ in $G'$ deterministically maintain the number \n of edges connecting $V_i$ to $V_j$. \n \\item \\textbf{Post-processing:} Compute a minimum vertex cover $I'$ in the multi-graph $G'$. \n \\item \\textbf{Return} $I = \\cup_{V_j \\in I'} V_j$ as the vertex cover in $G$. \n\\end{enumerate}\n\n\\noindent \\textbf{Analysis:} Regarding space, the dominating space requirement is the maintenance of the number\nof edges between every pair $V_i, V_j$. Since there are $n^{2-2\\epsilon}$ such pairs, this requires space $\\mathrm{O}(n^{2-2\\epsilon} \\cdot \\log n)$.\n\nConcerning the approximation factor, let $I^*$ be a minimum vertex cover in $G$. \nRecall that $I'$ is an optimal cover in $G'$ and hence $|I'| \\le |I^*|$ (edge contractions cannot increase the size of \na minimum vertex cover). \nSince every set $V_j$ is of size $n^\\epsilon$, \nthe computed vertex cover $I$ is of size at most $|I'| \\cdot n^{\\epsilon} \\le |I^*|n^{\\epsilon}$, which proves the approximation factor.\nBy construction of the algorithm, every edge is covered.\n\n\\begin{theorem} \\label{thm:vc-algorithm}\n There is a deterministic $n^\\epsilon$-approximation insertion-deletion streaming algorithm for \\textsf{Minimum Vertex Cover}\n that uses space $\\mathrm{O}(n^{2-2\\epsilon} \\log n)$.\n\\end{theorem}\n\n \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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\\vrule}}\n\\def\\color[rgb]{0.2,0.6,0.1}{\\color[rgb]{0.2,0.6,0.1}}\n\\newcommand{\\color{blue}}{\\color{blue}}\n\\newcommand{\\color{red}}{\\color{red}}\n\\newcommand{\\color[rgb]{0.2,0.6,0.1}}{\\color[rgb]{0.2,0.6,0.1}}\n\\newcommand{\\eq}[1]{{\\begin{equation}#1\\end{equation}}}\n\\newcommand{\\spl}[1]{{\\begin{aligned}#1\\end{aligned}}}\n\\newcommand{\\eqn}[1]{\\begin{equation*}#1\\end{equation*}}\n\\newcommand{\\eqsp}[1]{{\\begin{equation}\\begin{aligned}#1\\end{aligned}\\end{\nequation}}}\n\\newcommand{\\quot}[1]{\\begin{itemize}\\item[]``#1\"\\end{itemize}}\n\n\\mathsurround 1pt\n\n\\newcommand{\\red}[1]{{\\color{red}#1}}\n\n\n\\title[Well-posedness of Hardy-H\\'enon equation]{Optimal well-posedness and forward self-similar solution for the Hardy-H\\'enon parabolic equation in critical weighted Lebesgue spaces}\n\n\\date\\today\n\n\\author[N. Chikami, M. Ikeda and K. Taniguchi]{Noboru Chikami, Masahiro Ikeda and Koichi Taniguchi}\n\\address[N. Chikami]\n\nGraduate School of Engineering, \nNagoya Institute of Technology, \nGokiso-cho, Showa-ku, Nagoya \n466-8555, Japan.}\n\\email{chikami.noboru@nitech.ac.jp}\n\\address[M. Ikeda]\n{Faculty of Science and Technology,\nKeio University, \n3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan\/ Center for Advanced Intelligence Project\nRIKEN, Japan.}\n\\email{masahiro.ikeda@keio.jp\/masahiro.ikeda@riken.jp}\n\\address[K. Taniguchi]\n{Advanced Institute for Materials Research,\nTohoku University,\n2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan.}\n\\email{koichi.taniguchi.b7@tohoku.ac.jp}\n\n\\keywords{Hardy-H\\'enon parabolic equation, well-posedness, global existence, nonexistence, self-similar solution}\n\n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\n\\begin{document}\n\n\\footnote[0]\n{2010 {\\it Mathematics Subject Classification.}\nPrimary 35K05; Secondary 35B40;}\n\\maketitle\n\n\\begin{abstract}\nThe Cauchy problem for the Hardy-H\\'enon parabolic equation is studied in the critical and subcritical regime in weighted Lebesgue spaces on the Euclidean space $\\mathbb{R}^d$. \nWell-posedness for singular initial data and existence of non-radial forward self-similar solution of the problem are previously shown only for the Hardy and Fujita cases ($\\gamma\\le 0$) in earlier works. \nThe weighted spaces enable us to treat the potential $|x|^{\\gamma}$ as an increase or decrease of the weight, thereby we can prove well-posedness to the problem for all $\\gamma$ with $-\\min\\{2,d\\}<\\gamma$ including the H\\'enon case ($\\gamma>0$). \nAs a byproduct of the well-posedness, the self-similar solutions to the problem are also constructed for all $\\gamma$ without restrictions. \nA non-existence result of local solution for supercritical data is also shown. \nTherefore our critical exponent $s_c$ turns out to be optimal in regards to the solvability. \n\\end{abstract} \n\n\n\\section{Introduction}\\label{sec:1}\n\n\\subsection{Background and setting of the problem}\nWe consider the Cauchy problem of the Hardy-H\\'enon parabolic equation\n\\begin{equation}\\label{HH}\n\t\\begin{cases}\n\t\t\\partial_t u - \\Delta u = |\\cdot|^{\\gamma} |u|^{\\alpha-1} u,\n\t\t\t&(t,x)\\in (0,T)\\times D, \\\\\n\t\tu(0) = u_0 \\in L^q_{s}(\\mathbb{R}^d),\n\t\\end{cases}\n\\end{equation}\nwhere  $T>0,$ $d\\in \\mathbb{N}$, $\\gamma\\in \\mathbb{R},$ $\\alpha\\in \\mathbb{R},$ \n$D:=\\mathbb{R}^d$ if $\\gamma\\ge0$ and $D:=\\mathbb{R}^d\\setminus\\{0\\}$ if $\\gamma<0.$ \nHere, $\\partial_t:=\\partial\/\\partial t$ is the time derivative, \n$\\Delta:=\\sum_{j=1}^d\\partial^2\/\\partial x_j^2$ is the Laplace operator on $\\mathbb{R}^d$, \n$u=u(t,x)$ is the unknown real- or complex-valued function on \n$(0,T)\\times \\mathbb R^d$, and $u_0=u_0(x)$ is a prescribed real- or \ncomplex-valued function on $\\mathbb R^d$. \nIn this paper, we assume that the initial data $u_0$ belongs to weighted Lebesgue spaces $L^q_s(\\mathbb{R}^d)$ given by\n\\[L^q_s(\\mathbb{R}^d):=\\left\\{ f \\in \\mathcal{M} (\\mathbb{R}^d) \\,;\\, \n\t\\|f\\|_{L^q_s}\n\t< \\infty \\right\\}\n\\] endowed with the norm\n\\[\n\t\\|f\\|_{L^q_s} := \\left(\\int_{\\mathbb{R}^d} ( |x|^s |f(x)|)^q \\, dx \\right)^\\frac1{q},\n\\]\nwhere $s\\in \\mathbb{R}$ and $q\\in [1,\\infty]$ and $\\mathcal{M} (\\mathbb{R}^d)$ denotes the set of all\nLebesgue measurable functions on $\\mathbb{R}^d$. \nWe express the time-space-dependent function \n$u$ as $u(t)$ or $u(t,x)$ depending on circumstances. \nWe introduce a exponent $\\alpha_F(d,\\gamma)$ given by\n\\[\n    \\alpha_F(d,\\gamma):=1+\\frac{2+\\gamma}{d},\n\\]\nwhich is often referred as the {\\it Fujita exponent} and is known to divide the existence and \nnonexistence of positive global solutions (See \\cite[Theorem 1.6]{Qi1998}). \n\nThe equation \\eqref{HH} with $\\gamma<0$ is known as a \n{\\it Hardy parabolic equation} while that with $\\gamma>0$ \nis known as a {\\it H\\'enon parabolic equation}. \nThe elliptic part of \\eqref{HH}, that is, \n\\begin{equation}\\nonumbe\n    -\\Delta \\phi=|x|^{\\gamma}|\\phi|^{\\alpha-1}\\phi,\\ \\ \\ x\\in \\mathbb{R}^d,\n\\end{equation}\nwas proposed by H\\'enon as a model to study the rotating \nstellar systems (see \\cite{H-1973}), \nand has been extensively studied in the mathematical context, \nespecially in the field of nonlinear analysis and variational methods \n(see \\cite{GhoMor2013} for example). \nThe case $\\gamma=0$ corresponds to a heat equation with \na standard power-type nonlinearity, often called the {\\it Fujita equation}, \nwhich has been extensively studied in various directions. \nRegarding well-posedness of the Fujita equation ($\\gamma=0$) in Lebesgue \nspaces, we refer to \\cites{Wei1979, Wei1980, Gig86}, among many. \nConcerning the global dynamics and asymptotic behaviors, \nwe refer to \\cites{Ish2008,IT-arxiv,CIT-arxiv} for the Fujita \nand Hardy cases of \\eqref{HH} with Sobolev-critical exponents. \nArticles \\cites{HisIsh2018, HisTak-arxiv} give definitive results on \nthe optimal singularity of initial data to assure the solvability for $\\gamma\\le 0.$ \nIn \\cite{Tay2020}, unconditional uniqueness has been established for the Hardy case $\\gamma<0.$ Concerning earlier conditional uniqueness when $\\gamma<0$, \nwe refer to \\cites{BenTayWei2017, Ben2019}. \nLastly, we refer to \\cite{Maj-arxiv} for the analysis of the problem \n\\eqref{HH} with an external forcing term in addition to the nonlinear term. \n\nLet us recall that the equation \\eqref{HH} is invariant under the scale transformation \n\\begin{equation}\\label{scale}\nu_{\\lambda}(t,x) \n:= \\lambda^{\\frac{2+\\gamma}{\\alpha-1}} u(\\lambda^2 t, \\lambda x), \n\t\\quad \\lambda>0.\n\\end{equation}\nMore precisely, if $u$ is the classical solution to \\eqref{HH}, \nthen $u_{\\lambda}$ defined as above also solves the equation \nwith the rescaled initial data $ \\lambda^{\\frac{2+\\gamma}{\\alpha-1}} u_0(\\lambda x).$\nUnder \\eqref{scale}, the $L^q_{s}(\\mathbb{R}^d)$-norm scales as follows:\n$\\|u_\\lambda (0)\\|_{L^q_{s}} \n\t= \\lambda^{-s+\\frac{2+\\gamma}{\\alpha-1}-\\frac{d}{q}} \\|u(0)\\|_{L^q_{s}}.$ \nWe say that the space $L^q_{s}(\\mathbb{R}^d)$ is (scale-){\\sl critical} if $s=s_c$ with \n\\begin{equation}\\label{d:sc}\n\ts_c=s_c(q)= s_c(d,\\gamma,\\alpha,q) := \\frac{2+\\gamma}{\\alpha-1} - \\frac{d}{q}, \n\\end{equation}\n{\\sl subcritical} if $s<s_c,$ and {\\sl supercritical} if $s>s_c.$ \nIn particular, when $s=s_c = 0,$ $L^{\\frac{d(\\alpha-1)}{2+\\gamma}}(\\mathbb{R}^d)$ \nis a critical Lebesgue space. \n\nOne of our purposes in this article is to establish well-posedness results in the critical and subcritical cases ($s\\le s_c$) for all the range of the parameter $\\gamma$ such that \n$-\\min\\{2,d\\} < \\gamma,$ including the H\\'enon case ($\\gamma>0$). \nIn terms of well-posedness in function spaces containing sign-changing singular data, \nthe equation \\eqref{HH} has been studied mainly for $\\gamma<0$ (Hardy case). \nAs far as we know, there has been no result concerning well-posedness in the sense of Hadamard (Existence, uniqueness and continuous dependency) of the H\\'enon parabolic equation $\\gamma>0$ for sign-changing singular data. \nFor the Hardy and Fujita cases that are well-studied, our results provide \nwell-posedness in new function spaces (See Remark \\ref{r:HH.LWP}). \nWe stress that the use of weighted spaces enables us to treat the equations \nfor all $\\gamma$ in a unified manner. \n\nOur second purpose of this article is to \nprove the existence of forward self-similar solutions for all of \nHardy, Fujita and H\\'enon cases, without restrictions on the exponent $\\alpha.$ \nA forward self-similar solution is a solution such that $u_{\\lambda} = u$ for all $\\lambda>0,$ \nwhere $u_{\\lambda}$ is as in \\eqref{scale}. \nIn \\cite[Lemma 4.4]{Wan1993}, \nthe existence of radially symmetric self-similar solutions for \n$d\\ge3$, $\\gamma>-2$ and $\\alpha\\ge1+\\frac{2(2+\\gamma)}{d-2}$ \nis established. \nLater, the case $\\alpha_F(d,\\gamma)<\\alpha<1+\\frac{2(2+\\gamma)}{d-2}$ \nis treated in \\cite{Hir2008} under some additional restriction on $\\gamma,$ \nnamely $\\gamma\\le 0$ for $d\\ge4$ and $\\gamma\\le \\sqrt{3}-1$ for $d=3.$   \nIn \\cite[Theorem 1.4]{BenTayWei2017}, the existence of self-similar solutions that are not \nnecessarily radially symmetric has been proved for all $\\alpha>\\alpha_F(d,\\gamma),$ \nbut only for the Hardy case $\\gamma<0$ (See also \\cite{Chi2019}). \nOur result (Theorem \\ref{t:HH.self.sim}) covers all the previous results and asserts the existence of non-radial forward self-similar solutions for $\\gamma$ and $\\alpha$ \nsuch that $-\\min(2,d)<\\gamma$ and $\\alpha > \\alpha_F(d,\\gamma)$. \n\nIn earlier works, the crux of the matter has been \nthe handling of the singular potential $|x|^{\\gamma}.$ \nIf $\\gamma<0$, the conventional methods are to regard the potential \n$|x|^{\\gamma}$ as a function belonging either to the Lorentz space $L^{\\frac{d}{-\\gamma},\\infty}(\\mathbb{R}^d)$ \n(\\cites{BenTayWei2017, Tay2020}) \nor the homogeneous Besov space $\\dot B^{\\frac{d}{q}+\\gamma}_{q,\\infty}(\\mathbb{R}^d),$ \n$1\\le q \\le \\infty$ (\\cite{Chi2019}), \nand apply appropriate versions of H\\\"older's inequality to establish suitable \nheat kernel estimates. In contrast to their previous works, in this article, \nwe treat the potential $|x|^{\\gamma}$ as the increase or decrease of the order of \nthe weight in $L^q_s(\\mathbb{R}^d)$-norms, thereby covering the H\\'enon case ($\\gamma>0$) as well. \nIn this regard, the introduction of the weighted spaces is crucial to our results. \nIndeed, if the data only belongs to the critical Lebesgue space, then we \nmay only treat the Hardy case ($\\gamma<0$) in our main theorem \n(See Remark \\ref{r:HH.LWP} below). \nThe proofs of the well-posedness results rely on Banach's fixed point theorem. \nThe essential ingredient in the proof of various nonlinear estimates is the following linear estimate for the heat semigroup $\\{e^{t\\Delta}\\}_{t>0}$ on weighted Lebesgue spaces: \n\\begin{equation}\\nonumber\n\t\\| e^{t\\Delta} f\\|_{L^q_{s'}} \n\t\\le C t^{-\\frac{d}2 (\\frac1{p}-\\frac1{q}) - \\frac{s-s'}{2} } \t\n\t\t\\| f\\|_{L^p_{s}},\n\\end{equation}\n(see Lemma \\ref{l:wLpLq} for precise statement), which is known in the literatures such as \\cite{Tsu2011} except for the end-point cases. \nIn this article, we first extend the above estimate to the end-point cases $(i)$ $1<p<q=\\infty,$ $(ii)$ $p=q=1,$ $(iii)$ $1=p<q<\\infty,$ \n$(iv)$ $p=q=\\infty$ and $(v)$ $(p,q)=(1,\\infty)$. \n\nTo complete the picture of the admissible range of our well-posedness results, we also discuss the non-existence of positive distributional local solutions to \\eqref{HH} \nfor suitable supercritical data $u_0 \\in L^q_{s}(\\mathbb{R}^d)$ with $s>s_c.$ \n\n\\subsection{Main results}\nIn order to state our results, we introduce the following auxiliary function spaces. Let $\\mathscr{D}'([0,T)\\times\\mathbb{R}^d)$ be the space of distributions on $[0,T)\\times\\mathbb{R}^d$. \n\\begin{definition}[Kato class]\n\\label{def:Kato}\nLet $T \\in (0,\\infty],$ $s\\in\\mathbb{R}$ and $q\\in [1,\\infty].$ \n\\begin{enumerate}[(1)]\n\\item In the critical regime, i.e. $\\tilde s=s_c$, where $s_c$ is defined by \\eqref{d:sc}, for $s<\\tilde s$, the space $\\mathcal{K}^{s}(T)$ is defined by \n\\begin{equation}\\nonumber\n   \\mathcal{K}^{s}(T)\n   :=\\left\\{u\\in \\mathscr{D}'([0,T)\\times\\mathbb{R}^d) \\,;\\, \n   \t\\|u\\|_{\\mathcal{K}^{s}(T')}\n   <\\infty\\ \\text{for any } T' \\in (0,T)\\right\\}\n\\end{equation}\nendowed with a norm\n\\[\n\\|u\\|_{\\mathcal K^{s}(T)}\n\t:=\\sup_{0\\le t\\le T}t^{\\frac{s_c -s}{2}} \\|u(t)\\|_{L^q_s}.\n\\]\nWe simply write $\\mathcal{K}^{s}=\\mathcal{K}^{s}(\\infty)$ \nwhen $T=\\infty,$ if it does not cause confusion. \n\\item In the subcritical regime, i.e. $\\tilde s<s_c$, for $s<\\tilde s$, the space $\\tilde{\\mathcal{K}}^{s}(T)$ is defined by \n\\begin{equation}\\nonumber\n   \\tilde{\\mathcal{K}}^{s}(T)\n   :=\\left\\{u\\in \\mathscr{D}'([0,T)\\times\\mathbb{R}^d) \\,;\\, \n   \t\\|u\\|_{\\tilde{\\mathcal{K}}^{s}(T')}\n   <\\infty\\ \\text{for any } T' \\in (0,T)\\right\\}\n\\end{equation}\nendowed with a norm\n\\[\n\\|u\\|_{\\tilde{\\mathcal{K}}^{s}(T)}\n\t:=\\sup_{0\\le t\\le T}t^{\\frac{\\tilde s -s}{2}} \\|u(t)\\|_{L^q_s}.\n\\]\n\\end{enumerate}\n\\end{definition}\nFor $t\\in \\mathbb{R}_+$, we introduce the heat kernel $g_t:\\mathbb{R}^d\\rightarrow \\mathbb{R}_+$ given by\n\\begin{equation}\\label{d:h.krnl}\n\tg_t(x) := (4\\pi t)^{-\\frac{d}{2}} e^{-\\frac{|x|^2}{4t}}, \\ x \\in\\mathbb{R}^d. \n\\end{equation}\nWe denote by $\\{e^{t\\Delta}\\}_{t\n\\ge 0}$ the free heat semigroup defined by \n\\[\n(e^{t\\Delta} \\varphi) (x) := (g_t \\ast \\varphi) (x)\n\\]\nfor $\\varphi \\in L^1_{loc}(\\mathbb{R}^d),$ where $\\ast$ denotes the convolution with respect to the space variable. Let $\\mathcal{S}'(\\mathbb{R}^d)$ denotes the space of the Schwarz distributions.\nFor $\\varphi \\in \\mathcal{S}'(\\mathbb{R}^d)$, $e^{t\\Delta}\\varphi$ \nis defined by duality. \n\nIn what follows, we denote by $C_0^\\infty(\\mathbb{R}^d)$ the space of all smooth functions \nwith compact support. We also denote by $\\mathcal{L}^q_s (\\mathbb{R}^d)$ \nthe closure of $C_0^\\infty(\\mathbb{R}^d)$ with respect to the topology of $L^q_s (\\mathbb{R}^d).$ \nNext we give a definition of mild solution as follows. \n\\begin{definition}[Mild solution]\\label{def:sol-A}\nLet $T \\in (0,\\infty]$, $\\tilde s\\le s_c$ and $u_0 \\in L^q_{\\tilde s} (\\mathbb{R}^d)$. Let $Y := \\mathcal{K}^s(T)$ if $\\tilde s = s_c$ \nand $Y := \\tilde{\\mathcal{K}}^{s}(T)$ if $\\tilde s<s_c.$\nA function $u : [0,T] \\times \\mathbb{R}^d \\to \\mathbb C\\ \\text{or}\\ \\mathbb{R}$ is called an \n$L^q_{\\tilde s} (\\mathbb{R}^d)$-mild solution to \\eqref{HH} with initial data \n$u(0)=u_0$ if it satisfies \n$u\\in C([0,T]; L^q_{\\tilde s} (\\mathbb{R}^d)) \\cap Y$\nand the integral equation \n\\begin{equation}\\label{integral-eq}\nu(t,x) = e^{t\\Delta} u_0(x) \n\t+ \\int_0^t e^{(t-\\tau)\\Delta} \n\t\\left\\{ |\\cdot|^{\\gamma} |u(\\tau,\\cdot)|^{\\alpha-1}u(\\tau, \\cdot)\\right\\}(x) \\, d\\tau\n\\end{equation}\nfor any $t \\in [0,T]$ and almost everywhere $x \\in \\mathbb{R}^d$.\nThe time $T$ is said to be the maximal existence time, which is denoted \nby $T_m$, if the solution cannot be extended beyond $[0,T).$ \nMore precisely, \n\\begin{equation}\\label{d:Tm}\n\tT_m = T_m (u_0) := \\sup \\left\\{T>0 \\,;\\, \n\t\t\\left.\\begin{aligned}&\\text{There exists } \\text{ a unique solution $u$ of \\eqref{HH}} \\\\\n\t\t\t&\\text{in } C([0,T]; L^q_{\\tilde s}(\\mathbb{R}^d)) \\cap Y \n\t\t\t\\text{ with initial data $u_0$}\n\t\t\\end{aligned}\\right. \\right\\}.\n\\end{equation}\nWe say that $u$ is global in time if $T_m = + \\infty$ \nand that $u$ blows up in a finite time otherwise. \nMoreover, we say that $u$ is dissipative if $T_m = + \\infty$ and \n\\[\n\t\\lim_{t\\to\\infty} \\|u(t)\\|_{L^q_{\\tilde s}} = 0. \n\\]\n\\end{definition}\nThe following is one of our main results on local well-posedness of \\eqref{HH} in the critical space $L^q_{s_c}(\\mathbb{R}^d)$. \n\\begin{thm}[Well-posedness in the critical space]\t\\label{t:HH.LWP}\nLet $d\\in\\mathbb{N},$ $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \n\\begin{equation}\\label{t:HH.LWP.c0}\n\t\\gamma> -\\min(2,d)\n\t\t\\quad\\text{and}\\quad\n\t\\alpha> \\alpha_F(d,\\gamma).\n\\end{equation}\nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{t:HH.LWP.c1}\n\t\\alpha\\le q \\le \\infty \n\t\t\\quad\\text{and}\\quad\n\t\\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\,\n\t\t\\frac{2}{d(\\alpha-1)} + \\frac{(d-2)\\alpha - d -\\gamma}{d(\\alpha-1)^2}\\right\\}\n\\end{equation}\nand let $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{t:HH.LWP.c2}\n\ts_c - \\frac{d(\\alpha-1)}{\\alpha} \\left(\\frac{2}{d(\\alpha-1)} - \\frac1{q} \\right) \\le s\n\t< \\min \\left\\{ s_c, \\, s_c + \\frac{(d-2)\\alpha - d -\\gamma}{\\alpha(\\alpha-1)} \\right\\}. \n\\end{equation}\nThen the Cauchy problem \\eqref{HH} is locally well-posed in $L^q_{s_c}(\\mathbb{R}^d)$ for arbitrary data $u_0\\in L^q_{s_c}(\\mathbb{R}^d)$ and globally well-posed for small data $u_0\\in L^q_{s_c}(\\mathbb{R}^d)$. \nMore precisely, the following assertions hold. \n\\begin{enumerate}[$(i)$]\n\\item {\\rm (}Existence{\\rm )} \nFor any $u_0 \\in L^q_{s_c}(\\mathbb{R}^d)$ with $q <\\infty$ \n(Replace $L^\\infty_{s_c}(\\mathbb{R}^d)$ with \n$\\mathcal{L}^\\infty_{s_c}(\\mathbb{R}^d)$ when $q = \\infty$), \nthere exist a positive number $T$ and an $L^q_{s_c}(\\mathbb{R}^d)$-mild solution \n$u\n\\ to \\eqref{HH} satisfying \n\\begin{equation}\\label{t:HH.LWP.est} \n\t\\|u\\|_{\\mathcal{K}^s(T)} \n\t\t\\le 2 \\|e^{t\\Delta} u_0 \\|_{\\mathcal{K}^s(T)}. \n\\end{equation}\nMoreover, the solution can be extended to the maximal interval $[0,T_m),$ \nwhere $T_m$ is defined by \\eqref{d:Tm}. \n\\item {\\rm (}Uniqueness{\\rm )} \nLet $T>0.$ If $u, v \\in \\mathcal{K}^s(T)$ satisfy \n\\eqref{integral-eq} with $u(0) = v(0)=u_0 \\in L^q_{s_c}(\\mathbb{R}^d)$ \n(Replace $L^\\infty_{s_c}(\\mathbb{R}^d)$ with $\\mathcal{L}^\\infty_{s_c}(\\mathbb{R}^d)$ when $q=\\infty$), \nthen $u=v$ on $[0,T].$\n\\item {\\rm (}Continuous dependence on initial data{\\rm )}  \nLet $u$ and $v$ be the $L^q_{s_c}(\\mathbb{R}^d)$-mild solutions constructed in (i) with given initial data $u_0$ and $v_0$ respectively. \nLet $T(u_0)$ and $T(v_0)$ be the corresponding existence times. \nThen there exists a constant $C$ depending on $u_0$ and $v_0$ such that \nthe solutions $u$ and $v$ satisfy \n\\begin{equation}\\nonumber\n\t\\|u-v\\|_{L^\\infty(0,T;L^q_{s_c}) \\cap \\mathcal{K}^s(T)} \n\t\\le C \\|u_0-v_0\\|_{L^q_{s_c}}\n\\end{equation}\nfor some $T\\le \\min\\{T(u_0), T(v_0)\\}.$ \n\\item {\\rm (}Blow-up criterion{\\rm )} \nIf $u$ is an $L^q_{s_c}(\\mathbb{R}^d)$-mild solution constructed in the assertion $(i)$ and \n$T_m<\\infty,$ then $\\|u\\|_{\\mathcal{K}^s(T_m)}=\\infty.$\n\\item {\\rm (}Small data global existence and dissipation{\\rm )} \nThere exists $\\varepsilon_0>0$ depending only on $d,\\gamma,\\alpha,q$ and $s$ such that if $u_0 \\in \\mathcal{S}'(\\mathbb{R}^d)$ satisfies \n$\\|e^{t\\Delta}u_0\\|_{\\mathcal{K}^s}<\\varepsilon_0,$ then $T_m=\\infty$ and \n$\\|u\\|_{\\mathcal{K}^s} \\le 2\\varepsilon_0.$ Moreover, the solution $u$ is dissipative. In particular, if $\\|u_0\\|_{L^p_{s_c}}$ is sufficiently small, then \n$\\|e^{t\\Delta}u_0\\|_{\\mathcal{K}^s}<\\varepsilon_0.$ \n\\end{enumerate}\n\\end{thm}\n\\begin{rem}[Optimality of the power $\\alpha$ for the nonlinearity]\nBy the blow-up result in \\cite{Qi1998}, \nthe condition $\\alpha>\\alpha_F(d,\\gamma)$ is known to be optimal. \nIndeed, if $\\alpha \\le \\alpha_F(d,\\gamma)$, then the solutions of \\eqref{HH}\nwith positive initial data blow up in a finite time. \n\\end{rem}\n\\begin{rem}[Uniqueness $(ii)$]\nIn $(ii)$, $T$ is arbitrary and there is no restriction on the size of the quantity $\\|u\\|_{\\mathcal{K}^s(T)}.$ We note that this uniqueness result concerns a so-called {\\sl conditional} uniqueness since we can prove that $u\\in \\mathcal{K}^s(T)$ is a solution to \\eqref{HH} if and only if \n$u \\in C([0,T] ; L^q_{s_c}(\\mathbb{R}^d)) \\cap \\mathcal{K}^s(T)$ is a solution to \\eqref{HH}, \nprovided that $u_0 \\in L^q_{s_c}(\\mathbb{R}^d).$ See Remark \\ref{r:crt.est2} below. \nWe note that for the Hardy case, unconditional uniqueness has been established by \\cite{Tay2020} in the Lebesgue framework. \n\\end{rem}\n\\begin{exa}[Small data global existence $(v)$]\nWe give a typical example of the initial data $u_0$ \nsatisfying the assumptions in $(v)$ : \n$u_0 \\in L^1_{loc}(\\mathbb{R}^d)$ such that \n$|u_0(x)| \\le c |x|^{-\\frac{2+\\gamma}{\\alpha-1}}$ for almost all $x\\in\\mathbb{R}^d,$ \nwhere $c$ is a sufficiently small constant. \nThis initial data in particular generates a self-similar solution. \nSee Theorem \\ref{t:HH.self.sim} below. \n\\end{exa}\n\\begin{rem}[New contributions for $\\gamma\\neq0$]\n\\label{r:HH.LWP}\nFor the Hardy case $\\gamma > 0,$ Theorem \\ref{t:HH.LWP} is new \nconcerning sign-changing solutions for singular initial data. \nTheorem \\ref{t:HH.LWP} also gives a new result in the Hardy case ($\\gamma<0$). \nIn particular, when $s_c \\equiv 0$, that is, $q = \\frac{d(\\alpha-1)}{2+\\gamma}$, the critical space is the usual Lebesgue space \n$L^{\\frac{d(\\alpha-1)}{2+\\gamma}}(\\mathbb{R}^d).$ \nTheorem \\ref{t:HH.LWP} gives a new well-posedness result in the usual Lebesgue space $L^{\\frac{d(\\alpha-1)}{2+\\gamma}}(\\mathbb{R}^d)$ for $d\\ge2$ and $-2<\\gamma<0$. \n\\end{rem}\n\\begin{rem}\nWe note that $s_c$ is always positive when $\\gamma>0$ \nwhile $s_c$ can be either negative or non-negative. In other words, \nthe initial data $u_0$ must have a stronger decay at infinity when $\\gamma>0.$ \n\\end{rem}\n\nWe next discuss global existence of forward self-similar solutions to \\eqref{HH}. \nAs mentioned earlier, the result below is not known in the literature for large $\\gamma>0.$ \n\\begin{thm}[Existence of forward self-similar solutions]\t\\label{t:HH.self.sim}\nLet $d\\in\\mathbb{N},$ $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \\eqref{t:HH.LWP.c0}. \nLet $\\varphi(x) := \\omega(x) |x|^{-\\frac{2+\\gamma}{\\alpha-1}},$ \nwhere $\\omega\\in L^\\infty(\\mathbb{R}^d)$ is homogeneous of degree 0 and \n$\\|\\omega\\|_{L^\\infty}$ is sufficiently small so that $\\|e^{t\\Delta}\\varphi\\|_{\\mathcal{K}^s}<\\varepsilon_0$, where $\\varepsilon_0$ appears in Theorem \\ref{t:HH.LWP}.\nThen there exists a self-similar solution $u_\\mathcal{S}$ of \n\\eqref{HH} with the initial data $\\varphi$ such that $u_\\mathcal{S}(t) \\to \\varphi$ in $\\mathcal{S}'(\\mathbb{R}^d)$ as $t\\to0.$ \n\\end{thm}\n\nThe following theorem deals with the local well-posedness \nof \\eqref{HH} in the subcritical space $L^q_{\\tilde s}(\\mathbb{R}^d)$ with $\\tilde s< s_c.$ \n\\begin{thm}[Well-posedness in the subcritical space]\t\\label{t:HH.LWP.sub}\nLet $d\\in\\mathbb{N},$ $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \\eqref{t:HH.LWP.c0}. \nLet $\\tilde s\\in\\mathbb{R}$ be such that \n\\begin{equation}\\label{t:HH.LWP.sub.cs}\n\t\\max\\left\\{-\\frac{d}{\\alpha}, \\, \\frac{\\gamma}{\\alpha-1} \\right\\}\n\t\t<\\tilde s < \\frac{2+\\gamma}{\\alpha-1}.\n\\end{equation}\nLet $q\\in[1,\\infty]$ be such that \n\\begin{equation}\\label{t:HH.LWP.sub.c1}\n\t\\alpha\\le q \\le \\infty \n\t\t\\quad\\text{and}\\quad\n\t-\\frac{\\tilde s}{d} < \\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\,\n\t\t\\frac1{\\alpha} \\left(1-\\frac{\\tilde s}{d} \\right), \\, \n\t\t\\frac1{d} \\left(\\frac{2 + \\gamma}{\\alpha-1} -\\tilde s \\right)  \\right\\}\n\\end{equation}\nand let $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{t:HH.LWP.sub.c2}\n\t\\frac{\\tilde s+\\gamma}{\\alpha} \\le s\n\t\t\\quad\\text{and}\\quad\n\t- \\frac{d}{q} < s \n\t< \\min \\left\\{ \\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}, \\tilde s \\right\\}. \n\\end{equation}\nThen the Cauchy problem \\eqref{HH} is locally well-posed in $L^q_{\\tilde{s}}(\\mathbb{R}^d)$ for arbitrary data $u_0\\in L^q_{\\tilde{s}}(\\mathbb{R}^d)$. More precisely, the following assertions hold. \n\\begin{enumerate}[$(i)$]\n\\item {\\rm (}Existence{\\rm )} \nFor any $u_0 \\in L^q_{\\tilde s}(\\mathbb{R}^d),$ \nthere exist a positive number $T$ depending only on $\\|u_0\\|_{L^q_{\\tilde s}}$ and an $L^q_{\\tilde s}(\\mathbb{R}^d)$-mild solution \n$u \n\\ to \\eqref{HH} satisfying \n\\begin{equation}\\nonumber\n\t\\|u\\|_{\\tilde{\\mathcal{K}}^s(T)} \n\t\t\\le 2 \\|e^{t\\Delta} u_0 \\|_{\\tilde{\\mathcal{K}}^s(T)}. \n\\end{equation}\nMoreover, the solution can be extended to the maximal interval \n$[0,T_m),$ where $T_m$ is defined by \\eqref{d:Tm}.\n\\item {\\rm (}Uniqueness in $\\tilde{\\mathcal{K}}^s(T)${\\rm )} \nLet $T>0.$ If $u, v \\in \\tilde{\\mathcal{K}}^s(T)$ satisfy \n\\eqref{integral-eq} with $u(0) = v(0)=u_0,$ then $u=v$ on $[0,T].$ \n\\item {\\rm (}Continuous dependence on initial data{\\rm )}  \nFor any initial data $u_0$ and $v_0$ in $L^q_{\\tilde s}(\\mathbb{R}^d),$ \nlet $T(u_0)$ and $T(v_0)$ be the corresponding existence time given by $(i).$ \nThen there exists a constant $C$ depending on $u_0$ and $v_0$ such that \nthe corresponding solutions $u$ and $v$ satisfy \n\\begin{equation}\\nonumber\n\t\\|u-v\\|_{L^\\infty(0,T;L^q_{\\tilde s}) \\cap \\tilde{\\mathcal{K}}^s(T)} \n\t\\le C \\|u_0-v_0\\|_{L^q_{\\tilde s}}\n\\end{equation}\nfor some $T\\le \\min\\{T(u_0), T(v_0)\\}.$ \n\\item {\\rm (}Blow-up criterion{\\rm )} If $T_m<\\infty,$ \n\tthen $\\lim_{t\\rightarrow T_m-0}\\|u(t)\\|_{L^q_{\\tilde s}}=\\infty.$ \nMoreover, the following lower bound of blow-up rate holds: \nthere exists a positive constant $C$ independent of $t$ such that \n\\begin{equation}\\label{t:HH.LWP:Tm}\n\t\\|u(t)\\|_{L^q_{\\tilde s}} \\ge  \\frac{C}{(T_m - t)^{\\frac{s_c-\\tilde s}{2}} }\n\\end{equation}\nfor $t\\in (0,T_m)$.\n\\end{enumerate}\n\\end{thm}\n\\begin{rem}\nNote that \n\\eqref{t:HH.LWP.sub.c1} implies $\\tilde s<s_c,$ i.e., $u_0 \\in L^q_{\\tilde s}(\\mathbb{R}^d)$ \nis a scale-subcritical data.  \n\\end{rem}\n\nFinally, for the scale-supercritical case, i.e. $s>s_c$, we prove non-existence of a weak local positive solution, whose definition is given below. More precisely, we may prove that there exists a positive initial data $u_0$ in $L^q_s(\\mathbb{R}^d)$ with $s>s_c$ that does not generate a \nlocal solution to \\eqref{HH} even in the distributional sense. \n\\begin{definition}[Weak solution]\n\\label{d:w.sol}\nLet $T>0$. We call a function $u:[0,T)\\times \\mathbb{R}^d\\rightarrow \\mathbb{R}$ a weak solution to the Cauchy problem \\eqref{HH} \nif $u$ belongs to $L^{\\alpha}(0,T;L^{\\alpha}_{\\frac{\\gamma}{\\alpha},loc}(\\mathbb{R}^d))$ \nand if it satisfies the equation \\eqref{HH} in the distributional sense, i.e., \n\\begin{align}\\label{weak}\n\\notag\\int_{\\mathbb{R}^d} &u(T',x) \\eta (T',x) \\, dx-\\int_{\\mathbb{R}^d} u_0(x) \\eta (0,x) \\, dx\\\\\n\t&= \\int_{[0,T']\\times\\mathbb{R}^d} u(t ,x)(\\Delta \\eta + \\eta_t) (t ,x) \n\t+ |x|^{\\gamma} |u(t, x)|^{\\alpha-1} u(t,x) \\,\\eta(t,x)  \\, dx\\,dt\n\\end{align}\nfor all $T'\\in [0,T]$ and for all $\\eta \\in C^{1,2}([0,T]\\times \\mathbb{R}^d)$ such that\n$\\operatorname{supp} \\eta(t, \\cdot)$ is compact. \n\\end{definition}\nWe remark that our $L^q_{\\tilde s}(\\mathbb{R}^d)$-mild solutions \nare weak solutions in the above sense. See Lemma \\ref{mildweak} in Appendix.\n\\begin{thm}[Nonexistence of local positive weak solution]\n\\label{t:nonex}\nLet $d\\in \\mathbb N$ and $\\gamma \\in \\mathbb R$. \nAssume that $q\\in [1,\\infty],$ $\\alpha\\in\\mathbb{R}$ and $s\\in\\mathbb{R}$ satisfy \n$\\alpha>\\max(1, \\alpha_F(d,\\gamma))$ and $s>s_c$. \nThen there exists an initial data $u_0 \\in L^q_s (\\mathbb{R}^d)$ such that the \nproblem \\eqref{HH} with $u(0)=u_0$ has no local positive weak solution. \n\\end{thm}\n\\bigbreak\nThe rest of the paper is organized as follows: In Section 2, we \nprove the linear estimates and nonlinear ones in weighted Lebesgue spaces. \nSection 3 is devoted to the proof of Theorems \\ref{t:HH.LWP}, \\ref{t:HH.LWP.sub} \nand \\ref{t:HH.self.sim}. \nWe then give a sketch of the proof of Theorem \\ref{t:nonex} in Section 5. \nIn Appendix, we collect some elementary properties related to our function spaces and prove Lemma \\ref{mildweak}. \n\n\\section{Linear and nonlinear estimates}\nThroughout the rest of the paper, we denote by $C$ \na harmless constant that may change from line to line. \n\\subsection{Linear estimate}\nThe following estimate for the heat semigroup $\\{e^{t\\Delta}\\}_{t\\ge0}$ in weighted Lebesgue space is known except for the endpoint cases (see \\cites{Tsu2011, OkaTsu2016}).\n\\begin{lem}[Linear estimate]\n\t\\label{l:wLpLq}\nLet $d\\in\\mathbb N,$ $1\\le p \\le q \\le \\infty$ and  \n\\begin{equation}\t\\label{l:wLpLq:cs}\n\t-\\frac{d}{q} < s' \\le s < d\\left( 1-\\frac{1}{p} \\right).\n\\end{equation}\nIn addition, $s\\le 0$ when $p=1$ and $0\\le s'$ when $q=\\infty.$ \nIn particular,  \\eqref{l:wLpLq:cs} is understood as $s'=s=0$ when $p=1$ and $q=\\infty.$ \nThen there exists some positive constant $C$ depending on $d,$ $p,$ $q,$ $s$ and\n$s'$ such that \n\\begin{equation}\\nonumber\n\t\\| e^{t\\Delta} f\\|_{L^q_{s'}} \n\t\\le C t^{-\\frac{d}2 (\\frac1{p}-\\frac1{q}) - \\frac{s-s'}{2} } \n\t\t\\| f\\|_{L^p_{s}}\n\\end{equation}\nfor all $f\\in L^p_{s}(\\mathbb{R}^d)$ and $t>0$. \nMoreover, condition \\eqref{l:wLpLq:cs} is optimal. \n\\end{lem}\nWe mainly focus on the endpoint cases in the following proof.\n\\begin{proof}\nThe inequality for $1< p\\le q <\\infty$ follows from Lemma 3.2, \\cite[Proposition C.1]{OkaTsu2016} and the fact that the weight function \n$|x|^{s p}$ belongs to the Muckenhoupt class $A_p$ if and only if \n$- \\frac{d}{p} < s < d(1- \\frac1{p}).$ \n\nFor the endpoint exponents, we divide the proof into five cases : \n$(i)$ $1<p<q=\\infty,$ $(ii)$ $p=q=1,$ $(iii)$ $1=p<q<\\infty,$ \n$(iv)$ $p=q=\\infty$ and $(v)$ $(p,q)=(1,\\infty).$ \nIt suffices to prove the inequality for $e^{\\Delta} f$ and then resort to \na dilation argument as in the proof of \\cite[Proposition 2.1]{BenTayWei2017}. \n\nThroughout the proof of this lemma, \nwe write $a\\lesssim b$ if $a \\le C b$ with some constant $C.$\\\\ \n\\underline{$(i)$ $1<p<q=\\infty$}: Since $|x|^{s'} \\lesssim |x-y|^{s'} + |y|^{s'}$ if $s' \\ge 0,$ we have \n\\begin{equation*}\n\t|x|^{s'} | e^{\\Delta} f(x)| \n\t\\lesssim \\int_{\\mathbb{R}^d} |x-y|^{s'} g(x-y) |f(x)| \\, dy \n\t\t+ \\int_{\\mathbb{R}^d} |y|^{s'} g(x-y) |f(x)| \\, dy = : I_1 + I_2.\n\\end{equation*}\nFor $I_1,$ H\\\"older's inequality with $\\frac1{p}+\\frac1{p'}=1,$ $p>1,$ leads to \n\\begin{equation*}\\nonumber\n\tI_1\n\t\t \\le \\left( \\int_{\\mathbb{R}^d} (|y|^{-s} |x-y|^{s'} g(x-y) )^{p'}\\, dy \\right)^{\\frac1{p'}}  \\, \\|f\\|_{L^p_s} \n\t\t\\lesssim \\|f\\|_{L^p_s}, \n\\end{equation*}\nthanks to Lemma \\ref{l:g.unfrm.bnd} $(1)$ with $q\\equiv p',$ $a\\equiv s$ \nand $b\\equiv s',$ where $0\\le s<\\frac{d}{p'}$ and $s'\\ge0.$ \nSimilarly, H\\\"older's inequality and Lemma \\ref{l:g.unfrm.bnd} $(2)$ \nwith $q\\equiv p'$ and $c\\equiv s-s'$ yields \n\\begin{align*}\\nonumber\n\tI_2\n\t\t& \\le \\left( \\int_{\\mathbb{R}^d} (|y|^{-(s-s')} g(x-y) )^{p'}\\, dy \\right)^{\\frac1{p'}}  \\, \\|f\\|_{L^p_s} \n\t\t\\lesssim \\|f\\|_{L^p_s},\n\\end{align*}\nwhere $0\\le s-s' < \\frac{d}{p'}.$ \nThus, $\\| e^{\\Delta} f\\|_{L^{\\infty}_{s'}} \\lesssim \\| f\\|_{L^p_{s}}$ provided that \n$0 \\le s' \\le s < d\\left(1-\\frac1{p}\\right).$ \n\n\\underline{$(ii)$ $p=q=1$}: We have \n$|y|^{-s} \\lesssim |x-y|^{-s} + |x|^{-s}$ if $s\\le 0$ and thus \n\\begin{align*}\n\t\\|e^{\\Delta} f\\|_{L^1_{s'}} \n\n\t&\\lesssim \\int_{\\mathbb{R}^d} |x|^{s'} \\int_{\\mathbb{R}^d}  g(x-y) |x-y|^{-s} |y|^{s} |f(y)| \\,dy \\, dx \\\\\n\t\t&\\qquad\\qquad+ \\int_{\\mathbb{R}^d} |x|^{s'-s} \\int_{\\mathbb{R}^d}  g(x-y) |y|^{s} |f(y)| \\,dy \\, dx \\\\ \n\t&\\lesssim \\int_{\\mathbb{R}^d} \\left( \\int_{\\mathbb{R}^d} |x|^{s'} g(x-y) |x-y|^{-s} \\, dx \\right) |y|^{s} |f(y)|  \\,dy\\\\\n\t\t&\\qquad\\qquad+ \\int_{\\mathbb{R}^d} \\left( \\int_{\\mathbb{R}^d}  |x|^{s'-s}  g(x-y) \\, dx\\right) |y|^{s} |f(y)| \\,dy \\ \\lesssim \\|f\\|_{L^1_s}\n\\end{align*}\nthanks to Fubini's theorem and Lemma \\ref{l:g.unfrm.bnd} with \n$q\\equiv 1,$ $a\\equiv -s',$ $b\\equiv -s$ and $c\\equiv s-s',$ where \n$0 \\le -s' < d,$ $0 \\le -s$ and $0 \\le s-s' < d.$ \nThus, $\\| e^{\\Delta} f\\|_{L^{1}_{s'}} \\lesssim \\| f\\|_{L^1_{s}}$ provided that $-d<s'\\le s \\le 0.$\n\n\\underline{$(iii)$ $1=p < q < \\infty$}: By H\\\"older's inequality \nwith $1=\\frac1{q}+\\frac1{q'},$ $q<\\infty,$ we have  \n\\begin{align*}\n\t|e^{\\Delta} f(x)| \n\n\t&\\le \\left( \\int_{\\mathbb{R}^d} |y|^{-sq} g(x-y)^q |y|^s |f(y)| \\, dy \\right)^{\\frac1{q}} \\|f\\|_{L^1_s}^{\\frac1{q'}}\n\\end{align*}\nfor $s\\le 0.$ Taking the $L^q_{s'}(\\mathbb{R}^d)$-norm of the both sides of the above, we obtain \n\\begin{align*}\n\t\\|e^{\\Delta} f\\|_{L^q_{s'}} \n\t&\\le \\left( \\int_{\\mathbb{R}^d} |x|^{s'q} \\left( \\int_{\\mathbb{R}^d} |y|^{s(1-q)} g(x-y)^q |f(y)| \\, dy \\right) dx \\right)^{\\frac1{q}} \\|f\\|_{L^1_s}^{\\frac1{q'}}.\n\\end{align*}\nSince $|y|^{-qs} \\lesssim |x-y|^{-qs} + |x|^{-qs}$ if $s<0,$ \nFubini's theorem and Lemma \\ref{l:g.unfrm.bnd} with \n$q\\equiv q,$ $a\\equiv -s',$ $b\\equiv -s$ and $c\\equiv s-s'$ yield  \n\\begin{align*}\n\\int_{\\mathbb{R}^d} |x|^{s'q} \n\t&\\left( \\int_{\\mathbb{R}^d} |y|^{-sq} g(x-y)^q |y|^s |f(y)| \\, dy \\right) dx \\\\\n\t&\\lesssim \\int_{\\mathbb{R}^d} |x|^{s'q} \\left( \\int_{\\mathbb{R}^d} |x-y|^{-qs} g(x-y)^q |y|^{s} |f(y)| \\, dy \\right) dx \\\\\n\t&\\qquad\\qquad \n\t+ \\int_{\\mathbb{R}^d} |x|^{-(s-s')q} \\left( \\int_{\\mathbb{R}^d} g(x-y)^q |y|^{s} |f(y)| \\, dy \\right) dx \\\\\n\t&\\lesssim \\int_{\\mathbb{R}^d} |y|^{s} |f(y)| \\left( \\int_{\\mathbb{R}^d} (|x|^{s'} |x-y|^{-s} g(x-y) )^q \\, dx \\right) dy \\\\\n\t&\\qquad\\qquad \n\t+ \\int_{\\mathbb{R}^d} |y|^{s} |f(y)| \\left( \\int_{\\mathbb{R}^d} (|x|^{-(s-s')} g(x-y))^q \\, dx \\right) dy\n\\lesssim \\|f\\|_{L^1_s}, \n\\end{align*}\nwhere $0\\le -s' < \\frac{d}{q},$ $0\\le -s$ and $0 \\le s-s' < \\frac{d}{q}.$ \nThus, $\\| e^{\\Delta} f\\|_{L^{q}_{s'}} \\lesssim \\| f\\|_{L^1_{s}}$ provided that \n$-\\frac{d}{q}<s'\\le s \\le 0.$\n\n\\underline{$(iv)$ $p=q=\\infty$}: \nSince $|x|^{s'} \\lesssim |x-y|^{s'} + |y|^{s'}$ if $s' \\ge 0,$ we have \n\\begin{align*}\n\t|x|^{s'} &|e^{\\Delta} f(x)| \n\t\\le |x|^{s'} \\int_{\\mathbb{R}^d} |y|^{-s} g(x-y) \\, dy \\|f\\|_{L^\\infty_s}\\\\\n\t&\\lesssim \\left(  \\int_{\\mathbb{R}^d} |y|^{-s} |x-y|^{s'} g(x-y) \\, dy \n\t\t+ \\int_{\\mathbb{R}^d} |y|^{s'-s} g(x-y) \\, dy \\right)  \\|f\\|_{L^\\infty_s}\n\t\\lesssim \\|f\\|_{L^\\infty_s}\n\\end{align*}\nthanks to Lemma \\ref{l:g.unfrm.bnd} with \n$q\\equiv 1,$ $a\\equiv s,$ $b\\equiv s'$ and $c\\equiv s-s',$ where \n$0\\le s< d,$ $0\\le s'$ and $0\\le s-s' < d.$ \nThus, $\\| e^{\\Delta} f\\|_{L^{\\infty}_{s'}} \\lesssim \\| f\\|_{L^{\\infty}_{s}}$ provided that \n$0\\le s'\\le s <d.$\nThe case $(v)$ $(p,q) = (1,\\infty)$ is trivial. \nWe complete the proof of the endpoint estimates. \n\n\\smallbreak\nNext, we prove the optimality of \\eqref{l:wLpLq:cs} for \n$1< p \\le q < \\infty$ by contradiction. \nSuppose that the inequality holds when $s' \\le -\\frac{d}{q}.$ \nWe notice that every function $g$ in $L^q_{s'}(\\mathbb{R}^d)$ must satisfy \n$\\displaystyle \\liminf_{|x|\\to0} |x|^{\\frac{d}{q}+s'} |g(x)| =0$ thanks to \nCorollary \\ref{c:wLp.sg.dcy} in Appendix. \nIn particular, we have \n$\n\t\\Liminf_{|x|\\to0} |g(x)| =0\n$\nas $0 \\le -s'-\\frac{d}{q}.$ Since $0<\\frac{d}{p}+s < d,$ a function $f$ defined by \n\\[\nf(x) := \\left\\{\\begin{aligned}\n\t&C, &&|x|\\le 1\\\\\n\t&0, &&\\text{else},\n\\end{aligned}\\right. \n\\]\nwhere $C$ is a positive constant, belongs to $L^p_s(\\mathbb{R}^d).$ However, clearly \n\\[\n\\displaystyle \\liminf_{|x|\\to0} |e^{t\\Delta} f(x)| \\neq 0,\n\\]\nwhich implies that $e^{t\\Delta} f \\notin L^{q}_{s'}(\\mathbb{R}^d)$ and leads to a contradiction. \n\nThe optimality of the upper bound of \\eqref{l:wLpLq:cs} is based on the \nfact that the space $L^p_s(\\mathbb{R}^d)$ contains functions that are not in $L^1_{loc}(\\mathbb{R}^d),$ \nif $d\\left( 1-\\frac{1}{p} \\right)< s.$ Let \n\\[\nf(x) := \\left\\{\\begin{aligned}\n\t&|x|^{-d}, &&|x|\\le 1\\\\\n\t&0, &&\\text{else},\n\\end{aligned}\\right. \n\\]\nso that it belongs to $L^p_s(\\mathbb{R}^d)$ as $p(d-s)<d$ \n(Note that the space $L^p_s(\\mathbb{R}^d)$ is defined for all measurable functions). \nA standard argument then shows that $e^{t\\Delta} f$ does not make sense for the \nfunction $f.$ Indeed, for every $t>0$ and every $x$ such that $|x|\\le 1,$ the estimates hold:\n\\begin{equation*}\n\te^{t\\Delta} f(x)\n\t=  (4\\pi t)^{-\\frac{d}2} \\int_{|y|\\le 1} e^{\\frac{-|x-y|^2}{4t}} |y|^{-d} \\, dy \n\t\\ge  (4\\pi t)^{-\\frac{d}2} \\int_{|y|\\le 1} e^{-\\frac{1}{t}} |y|^{-d} \\, dy = \\infty, \n\\end{equation*}\nwhere we have used $|x-y|\\le 2.$ \nThus $e^{t\\Delta}$ is not well-defined in $L^p_s(\\mathbb{R}^d)$ if $d\\left( 1-\\frac{1}{p} \\right)< s.$ \nWhen $s= d(1-\\frac1{p}),$ it suffices to take \n\\[\nf(x) := \\left\\{\\begin{aligned}\n\t&|x|^{-d} \n\t\\left( \\log\\left(e+ \\frac1{|x|}\\right) \\right)^{-\\frac{a}{p}}, &&|x|\\le 1,\\\\\n\t&0, &&\\text{else},\n\\end{aligned}\\right. \n\\]\nwhere $p\\ge a>1,$ and show that $e^{t\\Delta} f$ is not well-defined for the function \nby carrying out the same argument as above. Thus, we conclude the lemma. \n\\end{proof}\n\n\n\\subsection{Nonlinear estimates}\nGiven $u_0\\in L^q_{s_c}(\\mathbb{R}^d)$ in the critical regime (resp. $L^q_{\\tilde{s}}(\\mathbb{R}^d)$ in the subcritical regime) and $T>0,$ let us define a map \n$\\Phi : u \\mapsto \\Phi(u)$ on $\\mathcal{K}^s(T)$ (resp. $\\tilde{\\mathcal{K}}^s(T)$) by \n\\begin{equation}\\label{map}\n\t\\Phi(u) (t) := e^{t\\Delta} u_0 + N(u)(t)\n\\end{equation}\nwith \n\\begin{equation}\\label{mapN}\n\tN(u)(t) :=  \\int_0^t e^{(t-\\tau)\\Delta} \n\t\\left\\{ |\\cdot|^{\\gamma} F(u(\\tau,\\cdot)) \\right\\} d\\tau\n\t\\quad\\text{and}\\quad\n\tF(u) := |u|^{\\alpha-1}u.\n\\end{equation}\n\n\\subsubsection{Critical case}\nThe following are the stability and contraction estimates in the critical regime. \nThe assertion $(2)$ below for $\\theta<1$ is not required in the proof of existence \nbut is used in the proof of uniqueness.  \n\\begin{lem\n\\label{l:Kato.est}\nLet $T \\in (0,\\infty]$ and $d\\in\\mathbb{N}.$ \nLet $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \\eqref{t:HH.LWP.c0}. \n\\begin{enumerate}[$(1)$]\n\\item \nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{l:Kato.est.c1}\n\t\\alpha\\le q \\le \\infty \\quad\\text{and}\\quad\n\t\\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\, \n\t\\frac{2}{d(\\alpha-1)} + \\frac{(d-2)\\alpha - d -\\gamma}{d\\alpha (\\alpha-1)} \\right\\}. \n\\end{equation}\nLet $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{l:Kato.est.c2}\n\t\\frac{\\gamma}{\\alpha-1}\\le s\n\t\t\\quad\\text{and}\\quad\n\t\\max\\left\\{- \\frac{d}{q}, \\, s_c - \\frac2{\\alpha} \\right\\} < s \n\t< \\min \\left\\{ s_c, \\, s_c + \\frac{(d-2)\\alpha - d -\\gamma}{\\alpha(\\alpha-1)} \\right\\},\n\\end{equation}\nwhere $s_c$ is as in \\eqref{d:sc}. \nThen there exists a positive constant $C_0$ \ndepending only on $d,$ $\\alpha,$ $\\gamma,$ $q$ and $s$ such that \nthe map $N$ defined by \\eqref{mapN} satisfies \n\\begin{equation}\\label{l:Kato.est1}\n\\|N(u)\\|_{\\mathcal{K}^s(T)} \n\t\\le C_0 \\|u\\|_{\\mathcal{K}^s(T)}^{\\alpha}\n\\end{equation}\nfor all $u \\in \\mathcal{K}^s(T).$ \n\\item\nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{l:Kato.est.c1'}\n\\begin{aligned}\n\t&\\alpha \\le q \\le \\infty, \\\\\n\t\t\\text{and}\\quad\n\t&\\frac1{q} < \\min \\left\\{ \\frac2{d(\\alpha-1)},\\, \n\t\\frac2{d(\\alpha-1)} + \\frac{\\theta(d-2)(\\alpha-1) - 2 - \\gamma}\n\t\t\t{d(\\alpha-1)(1+\\theta(\\alpha-1))} \\right\\},\n\\end{aligned}\n\\end{equation}\nwhere $\\theta \\in (0,1]$ ($\\frac1{2+\\gamma} < \\theta$ if d=1). \nLet $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{l:Kato.est.c2'}\n\\begin{aligned}\n&\ts_c - \\frac{d}{\\theta} \\left( \\frac{2}{d(\\alpha-1)} -\\frac{1}{q} \\right)\\le s \\\\\n\t\t\\text{and}\\quad\n&\t\\max\\left\\{ -\\frac{d}{q}, \\, s_c - \\frac2{1+\\theta(\\alpha-1)}  \\right\\}  \n\t< s < \\min\\left\\{ s_c, \\, s_c + \\frac{(d-2)\\alpha-d-2}{(1+\\theta(\\alpha-1))(\\alpha-1)}\n\t\t\t\t\t \\right\\}. \n\\end{aligned}\n\\end{equation}\nThen there exists a positive constant $C_1$ \ndepending only on $d,$ $\\alpha,$ $\\gamma,$ $q,$ $s$ and $\\theta$ such that \nthe map $N$ defined by \\eqref{mapN} satisfies  \n\\begin{equation}\\label{l:Kato.est2}\n\\begin{aligned}\n\\|N(u) - N(v)\\|_{\\mathcal{K}^s(T)} \n\t\\le C_1 &\\left(\\|u\\|_{\\mathcal{K}^s(T)}\n\t\t\t\t+\\|v\\|_{\\mathcal{K}^s(T)} \\right)^{\\theta(\\alpha-1)} \\\\\n\t\t&\\times\\left(\\|u\\|_{L^\\infty(0,T; L^q_{s_c}) }\n\t\t\t\t+\\|v\\|_{L^\\infty(0,T; L^q_{s_c}) } \\right)^{(1-\\theta)(\\alpha-1)} \n\t\\|u-v\\|_{\\mathcal{K}^s(T)}\n\\end{aligned}\n\\end{equation}\nfor all $u,v \\in \\mathcal{K}^s(T) \\cap L^\\infty(0,T ; L^q_{s_c}(\\mathbb{R}^d))$ \n($u,v \\in \\mathcal{K}^s(T)$ if $\\theta = 1$). \n\\end{enumerate}\n\\end{lem}\n\\begin{rem}\nNote that \\eqref{l:Kato.est.c1'} and \\eqref{l:Kato.est.c2'} for $\\theta=1$ \nare equivalent to \\eqref{l:Kato.est.c1} and \\eqref{l:Kato.est.c2}, respectively. \nThe estimate \\eqref{l:Kato.est2} fails for $\\theta=0$ as $C_1$ is divergent as \n$\\theta\\to0.$ \n\\end{rem}\n\\begin{proof}\nWe first prove \\eqref{l:Kato.est1}. We have \n\\begin{align*}\n\\|N(u)(t)\\|_{L^q_s} \n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 \\{(\\alpha-1)s - \\gamma\\}} \n\t\\| |\\cdot|^{\\gamma} F(u(\\tau)) \\|_{L^{\\frac{q}{\\alpha}}_{\\sigma}} d\\tau \\\\\n\\end{align*}\nby Lemma \\ref{l:wLpLq} with $q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s\\equiv s$ and $s'\\equiv \\sigma := \\alpha s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < s \\le \\alpha s -\\gamma < d(1-\\frac{\\alpha}{q}),$ i.e., \n\\begin{equation}\\label{l:Kato.est:pr1}\n\t\\alpha \\le q \\le \\infty, \\quad \n\t\\frac{\\gamma}{\\alpha-1} \\le s\n\t\t\\quad\\text{and}\\quad\n\t-\\frac{d}{q} < s < \\frac{\\gamma+d}{\\alpha}-\\frac{d}{q}. \n\\end{equation}\nAs $\\| |\\cdot|^{\\gamma} F(u) \\|_{L^{\\frac{q}{\\alpha}}_{\\sigma}}= \\| u \\|_{L^q_s}^{\\alpha}, $\nwe have \n\\begin{align*}\n\\|N(u)(t)\\|_{L^q_s} \n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 \\{(\\alpha-1)s - \\gamma\\}} \n\t\\tau^{-\\frac{(s_c-s)\\alpha}{2}} d\\tau \\times  \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha}, \n\\end{align*}\nwhere the last integral is bounded by \n\\begin{equation*}\n\tt^{-\\frac{s_c-s}2} B\\left(\\frac{\\alpha-1}2 (s_c -s), 1- \\frac{(s_c-s)\\alpha}2\\right),\n\\end{equation*}\nwhere $B:(0,\\infty)^2\\rightarrow \\mathbb{R}_{>0}$ is the beta function given by $B(x,y):=\\int_0^1t^{x-1}(1-t)^{y-1}dt$, which is convergent if and only if \n\\begin{equation}\\label{l:Kato.est:pr2}\n\ts_c - \\frac2{\\alpha} < s < s_c.\n\\end{equation}\nGathering \\eqref{l:Kato.est:pr1} and \\eqref{l:Kato.est:pr2}, \nwe have condition \\eqref{l:Kato.est.c2}. For such an $s$ to exist, \nit suffices to take $\\gamma,$ $\\alpha$ and $q$ so that \nconditions \\eqref{t:HH.LWP.c0}  and \\eqref{l:Kato.est.c1} are met. \n\n\t\\smallbreak\nWe next show \\eqref{l:Kato.est2}. \nSince there exists a constant $C=C(\\alpha)$ such that \n\\begin{equation}\\label{diff.pt.est}\n\t|F(u)-F(v)| \\le C (|u|^{\\alpha-1} + |v|^{\\alpha-1})|u-v| \n\t\\quad\\text{for all} \\quad u,v \\in \\mathbb{C},\n\\end{equation}\nwe have \n\\begin{align*}\n\\|N(u)(t) - N(v)(t)\\|_{L^q_s} \n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 \\{(\\alpha-1) (\\theta s + (1-\\theta) s_c) -\\gamma\\}} \\\\\n\t& \\quad \\times \\left\\| |\\cdot|^{\\gamma} \n\t(|u|^{\\alpha-1} + |v|^{\\alpha-1})|u-v| \\right\\|_{L^{\\frac{q}{\\alpha}}_{\\sigma}} \n\td\\tau,\n\\end{align*}\nthanks to Lemma \\ref{l:wLpLq} with $q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s\\equiv s$ and $s'\\equiv \\sigma :=  (\\alpha-1) (\\theta s + (1-\\theta) s_c) +s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < s \\le (\\alpha-1) (\\theta s + (1-\\theta) s_c) +s-\\gamma < d(1-\\frac{\\alpha}{q}),$ $\\theta \\in (0,1],$ i.e., \n\\begin{equation}\\label{l:Kato.est:pr1'}\n\\begin{aligned}\n&\t\\alpha \\le q \\le \\infty, \\quad \n\ts_c - \\frac{d}{\\theta} \\left( \\frac{2}{d(\\alpha-1)} -\\frac{1}{q} \\right) \\le s\\\\\n\t\t\\text{and}\\quad\n&\t-\\frac{d}{q} < s < s_c + \\frac1{1+\\theta(\\alpha-1)}\n\t\\left( d-2 -\\frac{2+\\gamma}{\\alpha-1} \\right). \n\\end{aligned}\n\\end{equation}\nBy H\\\"older's inequality with \n$\\frac{\\alpha}{q} = \\frac{\\theta(\\alpha-1)}{q} + \\frac{(1-\\theta)(\\alpha-1)}{q} + \\frac1{q},$  \nwe have \n\\begin{align*}\n& \\left\\| |\\cdot|^{\\gamma} \n\t(|u|^{\\alpha-1} + |v|^{\\alpha-1})|u-v| \\right\\|_{L^{\\frac{q}{\\alpha}}_{\\sigma}}  \\\\\n&\\le \\left( \\|u\\|_{L^q_s} + \\|v\\|_{L^q_s} \\right)^{\\theta(\\alpha-1)}\n\t\\left( \\|u\\|_{L^q_{s_c}} + \\|v\\|_{L^q_{s_c}} \\right)^{(1-\\theta)(\\alpha-1)} \n\t\t\\, \\|u-v\\|_{L^q_s}.\n\\end{align*}\nThus, \n\\begin{align*}\n&\\|N(u)(t) - N(v)(t)\\|_{L^q_s} \\\\\n&\\le C t^{-\\frac{s_c-s}2}\n\tB\\left(\\theta\\frac{\\alpha-1}2 (s_c -s), 1- \\frac{\\theta (\\alpha -1)+ 1}2 (s_c-s)\\right) \\\\\n&\\times \\left(\\|u\\|_{\\mathcal{K}^s(T)}\n\t\t\t\t+\\|v\\|_{\\mathcal{K}^s(T)} \\right)^{\\theta(\\alpha-1)} \n\t\t\\left(\\|u\\|_{L^\\infty(0,T; L^q_{s_c}) }\n\t\t\t\t+\\|v\\|_{L^\\infty(0,T; L^q_{s_c}) } \\right)^{(1-\\theta)(\\alpha-1)} \n \t\\|u-v\\|_{\\mathcal{K}^s(T)} \\\\\n\\end{align*}\nin which the last beta function is convergent if $\\theta>0$ and \n\\begin{equation}\\label{l:Kato.est:pr2'}\n\ts_c - \\frac2{\\theta(\\alpha-1)+1} < s < s_c.\n\\end{equation}\nGathering \\eqref{l:Kato.est:pr1'} and \\eqref{l:Kato.est:pr2'}, we deduce that \nthe restrictions for $s$ are \\eqref{l:Kato.est.c2'}. \nConsequently, for such an $s$ to exist, it suffices to take $q$ such that \\eqref{l:Kato.est.c1'}. \nFinally, for such a $q$ to exist, one must have \n$0<\\frac1{d(1+\\theta(\\alpha-1))} \\{ \\frac2{\\alpha-1} \n\t+ \\theta ( d- \\frac{2+\\gamma}{\\alpha-1} ) \\},$ i.e., \n$\\alpha>1+\\frac{2+\\gamma}{d} - \\frac2{\\theta d}$ and $0 < \\frac2{d(\\alpha-1)},$ \nboth of which hold thanks to \\eqref{t:HH.LWP.c0}. This concludes the proof of the lemma. \n\\end{proof}\n\nThe following is the stability estimate for the critical norm. \n\\begin{lem}\t\\label{l:crt.est}\nLet $T \\in (0,\\infty]$ and $d\\in\\mathbb{N}.$ \nLet $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \\eqref{t:HH.LWP.c0} . \nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{l:crt.est.c1}\n\t\\alpha\\le q \\le \\infty \n\t\t\\quad\\text{and}\\quad\n\t\\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\,\n\t\t\\frac{2}{d(\\alpha-1)} + \\frac{(d-2)\\alpha - d -\\gamma}{d(\\alpha-1)^2} \\right\\}\n\\end{equation}\nand let $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{l:crt.est.c2}\n\t s_c - \\frac{d(\\alpha-1)}{\\alpha} \\left(\\frac{2}{d(\\alpha-1)} - \\frac1{q} \\right) \\le s \n\t< \\min \\left\\{ s_c, \\, s_c + \\frac{(d-2)\\alpha - d -\\gamma}{\\alpha(\\alpha-1)} \\right\\}, \n\\end{equation}\nwhere $s_c$ is as in \\eqref{d:sc}. \nThen there exists a positive constant $C_2$ \ndepending only on $d,$ $\\alpha,$ $\\gamma,$ $q$ and $s$ such that \nthe map $N$ defined by \\eqref{mapN} satisfies \n\\begin{equation}\\nonumbe\n\t\\|N(u)\\|_{L^\\infty(0,T ; L^q_{s_c})} \n\t\t\\le C_2 \\|u\\|_{\\mathcal{K}^s(T)}^{\\alpha}\n\\end{equation}\nfor all $u,v \\in \\mathcal{K}^s(T).$ \n\\end{lem}\n\\begin{proof}\nLet $T>0$ and $u,v \\in \\mathcal{K}^s(T).$ We have \n\\begin{align*}\n\\|N(u)(t)\\|_{L^q_{s_c}} \n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 (\\alpha s -\\gamma- s_c)} \n\t \\| u(\\tau) \\|_{L^q_s}^{\\alpha} d\\tau \\\\\n&\\le C B\\left(\\frac{\\alpha}2 (s_c-s), 1 - \\frac{(s_c-s)\\alpha}2 \\right)  \n\t\\times  \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha}, \n\\end{align*}\nthanks to Lemma \\ref{l:wLpLq} with $q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s\\equiv s_c$ and $s'\\equiv \\alpha s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < s_c \\le \\alpha s -\\gamma < d(1-\\frac{\\alpha}{q}),$ i.e., \n\\begin{equation}\\nonumber\n\t-2<\\gamma, \\quad \n\t\\alpha \\le q \\le \\infty\t\\quad\\text{and}\\quad \n\t\\frac{s_c+\\gamma}{\\alpha} \\le s < \\frac{d+\\gamma}{\\alpha}-\\frac{d}{q}. \n\\end{equation}\nThe final beta function is convergent if \\eqref{l:Kato.est:pr2} holds. \nSince $s_c - \\frac2{\\alpha} \\le \\frac{s_c+\\gamma}{\\alpha},$\nthe restrictions on $s$ are \\eqref{l:crt.est.c2}. \nFor such an $s$ to exist, $q$ must satisfy \\eqref{l:crt.est.c1} in addition to \n$\\alpha\\le q \\le\\infty.$ \nIndeed, $\\frac{s_c + \\gamma}{\\alpha}<s_c$ \nis equivalent to $\\frac1{q} < \\frac2{d(\\alpha-1)}$ and \n$\\frac{s_c+\\gamma}{\\alpha} < s_c$ is equivalent to \n$\\frac1{q} < \\frac1{\\alpha-1} \\left(1-\\frac{2+\\gamma}{d(\\alpha-1)} \\right).$ \nThis completes the proof of the lemma. \n\\end{proof}\n\\begin{rem}\t\\label{r:crt.est2}\nNote that the above lemma along with Lemma \\ref{l:wLpLq} imply that \na solution $u \\in \\mathcal{K}^s(T)$ yields the regularity \n$u\\in C([0,T] ; L^q_{s_c}(\\mathbb{R}^d)),$ if $u_0 \\in L^{q}_{s_c}(\\mathbb{R}^d).$ Thus, \nif we allow the abuse of notation, the equivalence \n$\\mathcal{K}^s(T) = C([0,T] ; L^q_{s_c}(\\mathbb{R}^d)) \\cap \\mathcal{K}^s(T)$ \nholds as solution spaces of \\eqref{HH}. \n\\end{rem}\n\n\n\\subsubsection{Subcritical case}\nThe following are the stability and contraction estimates in the subcritical regime. \n\\begin{lem\n\\label{l:Kato.est.sub}\nLet $T \\in (0,\\infty]$ and $d\\in\\mathbb{N}.$ Let $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \n\\eqref{t:HH.LWP.c0}. \nFix $\\tilde s\\in\\mathbb{R}$ so that \n\\begin{equation}\\label{l:Kato.est.sub.c0}\n\t\\tilde s < \\frac{2+\\gamma}{\\alpha-1}.\n\\end{equation}\nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{l:Kato.est.sub.c1}\n\t\\alpha\\le q \\le \\infty\n\t\t\\quad\\text{and}\\quad\n\t\\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\,\n\t\t\\frac1{\\alpha} \\left(1 - \\frac{\\gamma}{d(\\alpha-1)}\\right), \\, \n\t\t\\frac1{d} \\left(\\frac{2+\\gamma}{\\alpha-1} -\\tilde s \\right)\\right\\}\n\\end{equation}\nand let $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{l:Kato.est.sub.c2}\n\t\\frac{\\gamma}{\\alpha-1}\\le s\n\t\t\\quad\\text{and}\\quad\n\t\\max\\left\\{\\tilde s - \\frac2{\\alpha}, \\, - \\frac{d}{q} \\right\\} < s \n\t< \\min \\left\\{ \\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}, \\, s_c \\right\\}, \n\\end{equation}\nwhere $s_c$ is as in \\eqref{d:sc}, \nThen there exist positive constants $\\tilde C_0$ and $\\tilde C_1$ \ndepending only on $d,$ $\\alpha,$ $\\gamma,$ $q,$ $\\tilde s$ and $s$ such that \nthe map $N$ defined by \\eqref{mapN} satisfies \n\\begin{equation}\\label{l:Kato.est.sub1}\n\\|N(u)\\|_{\\tilde{\\mathcal{K}}^s(T)} \n\t\\le \\tilde C_0 T^{\\frac{\\alpha-1}2(s_c-\\tilde s)} \\|u\\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha}\n\\end{equation}\nand \n\\begin{equation}\\label{l:Kato.est.sub2}\n\\|N(u) - N(v)\\|_{\\tilde{\\mathcal{K}}^s(T)} \n\t\\le \\tilde C_1 T^{\\frac{\\alpha-1}2(s_c-\\tilde s)} \\left( \\|u\\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha-1} + \\|v\\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha-1} \\right)\n\t\\|u-v\\|_{\\tilde{\\mathcal{K}}^s(T)}\n\\end{equation}\nfor all $u,v \\in \\tilde{\\mathcal{K}}^s(T).$ \n\\end{lem}\n\\begin{rem}\nNote that $\\frac1{q}<\\frac1{d} \\left(\\frac{2+\\gamma}{\\alpha-1} -\\tilde s \\right)$ \nin \\eqref{l:Kato.est.sub.c1} amounts to $\\tilde s<s_c,$ \nso the power of $T$ in \\eqref{l:Kato.est.sub1} and \\eqref{l:Kato.est.sub2} is positive. \n\\end{rem}\n\\begin{proof}[Proof of Lemma \\ref{l:Kato.est.sub}]\nWe have \n\\begin{align*}\n\\|N(u)(t)&\\|_{L^q_{s}} \n\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 ((\\alpha-1) s -\\gamma)} \n\t \\| u(\\tau) \\|_{L^q_s}^{\\alpha} d\\tau \\\\\n&\\le C t^{-\\frac12(s-\\tilde s)} t^{\\frac{\\alpha-1}2(s_c-\\tilde s)} B\\left(\\frac{(\\alpha-1)}2 (s_c-s), 1 - \\frac{(\\tilde s-s)\\alpha}2 \\right)  \n\t\\times  \\|u \\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha}, \n\\end{align*}\nthanks to Lemma \\ref{l:wLpLq} with\n$q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s\\equiv \\tilde s$ and $s'\\equiv \\alpha s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < s \\le \\alpha s -\\gamma < d(1-\\frac{\\alpha}{q}),$ i.e., \n\\begin{equation}\\nonumber\n\t\\alpha \\le q \\le \\infty, \\quad\n\t\\frac{\\gamma}{\\alpha-1} \\le s\t\\quad\\text{and}\\quad \n\t-\\frac{d}{q} < s < \\frac{d+\\gamma}{\\alpha}-\\frac{d}{q}. \n\\end{equation}\nThe final beta function is convergent if $\\tilde s-\\frac2{\\alpha}<s< s_c.$\nThus, the restrictions on $s$ are \\eqref{l:Kato.est.sub.c2}. \nFor such an $s$ to exist, $q$ must satisfy \\eqref{l:Kato.est.sub.c1}. \nFinally, for such a $q$ to exist, \nwe immediately see that $\\tilde s$ must satisfy \\eqref{l:Kato.est.sub.c0}. \n\nThe proof for the difference is similar to the above so we omit the details. \nThis completes the proof of the lemma. \n\\end{proof}\n\n\\begin{lem\n\\label{l:subcrt.est}\nLet $T \\in (0,\\infty]$ and $d\\in\\mathbb{N}.$ \nLet $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \\eqref{t:HH.LWP.c0}. \nFix $\\tilde s$ so that \\eqref{t:HH.LWP.sub.cs} is satisfied. \nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{l:subcrt.est.c1}\n\t\\alpha\\le q \\le \\infty\n\t\t\\quad\\text{and}\\quad\n\t-\\frac{\\tilde s}{d} <\\frac1{q} \n\t<\\min\\left\\{ \\frac1{\\alpha} \\left(1 -\\frac{\\tilde s}{d}\\right), \\, \n\t\t\t\\frac1{d} \\left(\\frac{2+\\gamma}{\\alpha-1} -\\tilde s \\right) \\right\\}\n\\end{equation}\nand let $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{l:subcrt.est.c2}\n\t\\frac{\\tilde s+\\gamma}{\\alpha} \\le s \n\t< \\min \\left\\{ \\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}, \\tilde s \\right\\}, \n\\end{equation}\nwhere $s_c$ is as in \\eqref{d:sc}. \nThen there exists a positive constant $\\tilde C_2$ \ndepending only on $d,$ $\\gamma,$ $\\alpha,$ $\\tilde s,$ $q$ and $s$ such that \nthe map $N$ defined by \\eqref{mapN} satisfies \n\\begin{equation}\\nonumber\n\t\\|N(u)\\|_{L^\\infty(0,T ; L^q_{\\tilde s})} \n\t\t\\le \\tilde C_2 T^{\\frac{\\alpha-1}2(s_c-\\tilde s)} \\|u\\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha}\n\\end{equation}\nfor all $u\\in \\tilde{\\mathcal{K}}^s(T).$ \n\\end{lem}\n\\begin{proof}\nWe have \n\\begin{align*}\n\\|N(u)(t)&\\|_{L^q_{\\tilde s}} \n\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 (\\alpha s -\\gamma- \\tilde s)} \n\t \\| u(\\tau) \\|_{L^q_s}^{\\alpha} d\\tau \\\\\n&\\le C t^{\\frac{\\alpha-1}2(s_c-\\tilde s)} B\\left(\\frac12 \\{(\\alpha-1)(s_c-s) + \\tilde s-s\\}, 1 - \\frac{(\\tilde s-s)\\alpha}2 \\right)  \n\t\\times  \\|u \\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha}, \n\\end{align*}\nthanks to Lemma \\ref{l:wLpLq} with\n$q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s\\equiv \\tilde s$ and $s'\\equiv \\alpha s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < \\tilde s \\le \\alpha s -\\gamma < d(1-\\frac{\\alpha}{q}),$ i.e., \n\\begin{equation}\\nonumber\n\t\\alpha \\le q \\le \\infty, \\quad\n\t-\\frac{\\tilde s}{d} <\\frac1{q}\t\t\\quad\\text{and}\\quad \n\t\\frac{\\tilde s+\\gamma}{\\alpha} \\le s < \\frac{d+\\gamma}{\\alpha}-\\frac{d}{q}. \n\\end{equation}\nThe final beta function is convergent if $\\tilde s-\\frac2{\\alpha}<s<\\tilde s< s_c.$\nSince $\\tilde s-\\frac2{\\alpha}<\\frac{\\tilde s+\\gamma}{\\alpha}$ \n(by $\\tilde s < s_c < \\frac{2+\\gamma}{\\alpha-1}$), the restrictions on $s$ are \n\\eqref{l:subcrt.est.c2}. \nFor such an $s$ to exist, $q$ must satisfy \\eqref{l:subcrt.est.c1} \nand $\\frac{\\gamma}{\\alpha-1} < \\tilde s.$ \nFinally, for such a $q$ to exist, $\\tilde s$ must satisfy \\eqref{t:HH.LWP.sub.cs}\nsince \n\\[\n\t-\\frac{d}{\\alpha-1}\n\t<\\max\\left\\{ -\\frac{d}{\\alpha}, \\, \\frac{\\gamma}{\\alpha-1} \\right\\} \\quad\\text{and}\\quad\n\t\\frac{2+\\gamma}{\\alpha-1} < d.\n\\]\nThis completes the proof of the lemma. \n\\end{proof}\n\n\\subsubsection{Upgrade of regularity}\nThe following lemma is used to show the regularity of the $L^q_{\\tilde{s}}(\\mathbb{R}^d)$-mild solution. \n\\begin{lem}\t\\label{l:b.strap}\nLet $p,q \\in [1,\\infty]$ and $s,s' \\in \\mathbb{R}.$ Under condition \\eqref{t:HH.LWP.c0}, \nlet pairs $(q,s)$ and $(p,s')$ be such that either\n\\begin{equation}\t\\label{l:b.strap:c}\n\\begin{aligned}\n\t&\\alpha \\le q < \\infty, \\quad\n\t\\max\\left\\{- \\frac{d}{q}, \\, \\frac{1}{\\alpha}\\left( \\gamma- \\frac{d}{q} \\right)\\right\\} < s \n\t<\\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}, \\\\\n\t&\\max\\left\\{0, \\, -\\frac{s}{d}, \\, \\frac{\\gamma-\\alpha s}{d} \\right\\}\n\t\t<\\frac1{p} \\le \\frac{1}{q}, \\quad\n\t -\\frac{d}{p} < s' \\le \\min\\{s, \\, \\alpha s -\\gamma\\}. \n\\end{aligned}\n\\end{equation}\nor \n\\begin{equation}\t\\label{l:b.strap:p=inf}\n\\begin{aligned}\n\t&\\alpha < q \\le \\infty, \\quad\n\t\\max\\left\\{0,\\, \\frac{\\gamma}{\\alpha}\\right\\} \\le s \n\t<\\frac{d+\\gamma}{\\alpha} - \\frac{d}{q},\\\\\n\t&p= \\infty, \\quad 0\\le s' \\le \\min\\{s, \\, \\alpha s -\\gamma\\}.\n\\end{aligned}\n\\end{equation}\nLet $u$ be the $L^q_{s_c}(\\mathbb{R}^d)$-mild solution of \\eqref{HH} with initial data $u_0 \\in \\mathcal{S}'(\\mathbb{R}^d)$ on $[0,T_m)$\nsuch that \n\\begin{equation}\\nonumber\n\t\\sup_{t\\in [0,T_m)} t^{\\frac{s_c(q)-s}{2} } \\|u(t)\\|_{L^q_{s}} < \\infty.\n\\end{equation}\nThen it follows that \n\\begin{equation}\\nonumber\n\t\\sup_{t\\in [0,T_m)} t^{\\frac{s_c(p)-s'}{2} } \\|u(t)\\|_{L^p_{s'}} < \\infty.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nWe use a similar argument as in \\cite{SnoTayWei2001} (See also \\cite{BenTayWei2017}). \nLet \n\\[\n\tA := \\sup_{t\\in [0,T_m)} t^{\\frac{s_c(q)-s}2}\\|u(t)\\|_{L^{q}_{s}}<\\infty.\n\\]\nLet $t\\in (0,T_m)$. We use the integral representation \n\\begin{equation}\\nonumber\n\tu(t) = e^{\\frac{t}2 \\Delta} u(t\/2) \n\t+ \\int_{\\frac{t}{2}}^t e^{(t-\\tau)\\Delta} \\left\\{ |\\cdot|^{\\gamma} \n\t\tF(u(\\tau,\\cdot)) \\right\\} d\\tau. \n\\end{equation}\nIt follows from Lemma \\ref{l:wLpLq} with $q\\equiv p,$ $p\\equiv q,$ \n$s'\\equiv s'$ and $s\\equiv s,$ that \n\\begin{align*}\n\t\\|e^{\\frac{t}2\\Delta} u(t\/2)\\|_{L^{p}_{s'}} \n\t\\le C t^{-\\frac{d}{2}(\\frac1{q}-\\frac1{p}) - \\frac{s-s'}2} \\|u(t\/2)\\|_{L^{q}_{s}} \n\t\\le C t^{-\\frac{1}{2}(\\frac{2+\\gamma}{\\alpha-1}-\\frac{d}{p} - s')} A \n\\end{align*}\nif \n\\begin{equation}\\label{l:b.strap:pr0}\n\\begin{aligned}\n\t&1\\le q \\le p \\le \\infty\n\t\t\\quad\\text{and}\\quad \n\t-\\frac{d}{p} < s' \\le s < d\\left(1-\\frac1{q}\\right) \\quad\\left(0\\le s' \\text{ if } p=\\infty\\right). \\\\\n\\end{aligned}\n\\end{equation}\nOn the other hand, \n\\begin{equation}\\nonumbe\n\\begin{aligned}\n\\|&N(u)(t)\\|_{L^{p}_{s'}} \n\\le  C \\int_{\\frac{t}{2}}^t \n\t\t(t-\\tau)^{-\\frac{d}2(\\frac{\\alpha}{q}-\\frac1{p}) - \\frac{\\alpha s -\\gamma-s'}2} \n\t\t \\| u(\\tau) \\|_{L^q_s}^{\\alpha} d\\tau \\\\\n&\\le CA^{\\alpha} \\int_{\\frac{t}{2}}^t (t-\\tau)^{-\\frac{d}2(\\frac{\\alpha}{q}-\\frac1{p}) - \\frac{\\alpha s -s'-\\gamma}2} \n\t\\tau^{-\\frac{(s_c-s)\\alpha}{2}} d\\tau \\\\\n&= CA^{\\alpha}  t^{-\\frac12(\\frac{2+\\gamma}{\\alpha-1} - \\frac{d}{p} - s')} \n \\int_{\\frac{1}{2}}^1 (1-\\tau)^{-\\frac{d}2(\\frac{\\alpha}{q}-\\frac1{p}) - \\frac{\\alpha s -s'-\\gamma}2} \n\t\\tau^{-\\frac{(s_c-s)\\alpha}{2}} d\\tau, \n\\end{aligned}\n\\end{equation}\nthanks to Lemma \\ref{l:wLpLq} with $q\\equiv p,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s'\\equiv s'$ and $s\\equiv \\alpha s-\\gamma,$ provided that \n\\begin{equation}\\label{l:b.strap:pr1}\n\\begin{aligned}\n\t&1\\le \\frac{q}{\\alpha} \\le p \\le \\infty\n\t\t\\quad\\text{and}\\quad \n\t-\\frac{d}{p} < s' \\le \\alpha s-\\gamma < d\\left(1-\\frac{\\alpha}{q}\\right)\n\t\\quad\\left(0\\le s' \\text{ if } p=\\infty\\right).\\\\\n\\end{aligned}\n\\end{equation}\ni.e., $\\alpha \\le q \\le \\infty,$ $\\frac1{p} \\le \\frac{\\alpha}{q},$\n$-\\frac{d}{p} < s' \\le \\alpha s-\\gamma$ ($0\\le s'\\le \\alpha s-\\gamma$ if $p=\\infty$) and $s < \\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}.$ \nThe final integral is convergent if \n\\begin{equation}\\label{l:b.strap:pr2}\n\t1-\\frac{d}2\\left(\\frac{\\alpha}{q}-\\frac1{p}\\right) - \\frac{\\alpha s-s' -\\gamma}2>0, \n\t\t\\quad\\text{i.e.,}\\quad\n\t\\alpha\\left( \\frac{d}{q} + s\\right) - 2 - \\gamma - \\frac{d}{p}<s'.\n\\end{equation}\nThus, we have \n\\begin{equation}\\nonumber\n\t\\sup_{t\\in [0,T_m)} t^{\\frac{s_c(p)-s'}{2} } \\|u(t)\\|_{L^p_{s'}} \n\t\\le C (A + A^{\\alpha})\n\\end{equation}\nunder \\eqref{l:b.strap:pr0}, \\eqref{l:b.strap:pr1} and \\eqref{l:b.strap:pr2}. \nSince $s < \\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}$ from \\eqref{l:b.strap:pr1}, we have \n$\\alpha( \\frac{d}{q} + s) - 2 - \\gamma - \\frac{d}{p} < - \\frac{d}{p}.$ Thus, \nthe conditions for $s'$ are that in \\eqref{l:b.strap:c}. \nBy tedious but straightforward computations, \nwe may easily see that under condition \\eqref{t:HH.LWP.c0}, \nthe necessary and sufficient conditions of \n\\eqref{l:b.strap:pr0}, \\eqref{l:b.strap:pr1} and \\eqref{l:b.strap:pr2} are \n\\eqref{l:b.strap:c} or \\eqref{l:b.strap:p=inf}. Hence, the lemma is proved. \n\\end{proof}\n\n\n\n\n\\section{Local well-posedness and self-similar solutions}\n\n\\subsection{Proof of Theorem \\ref{t:HH.LWP}}\nIn order to prove Theorem \\ref{t:HH.LWP}, we prepare the following lemma. \n\\begin{lem}\\label{l:exist.crt}\nLet positive numbers $\\rho>0$ and $M>0$ satisfy \n\\begin{equation}\\label{l:exist.crt.c0}\n\t\\rho + C_0 M^\\alpha \\le M \\quad\\text{and}\\quad\n\t2 C_1 M^{\\alpha-1} <1,\n\\end{equation}\nwhere $C_0$ and $C_1$ are as in Lemma \\ref{l:Kato.est}. \nUnder conditions \\eqref{t:HH.LWP.c0}, \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}, \nlet $T\\in (0,\\infty]$ and $u_0 \\in \\mathcal{S}'(\\mathbb{R}^d)$ be \nsuch that $e^{t\\Delta}u_0 \\in \\mathcal{K}^s(T).$ \nIf $\\|e^{t\\Delta}u_0\\|_{\\mathcal{K}^s(T)}\\le \\rho,$ \nthen a solution $u$ to \\eqref{HH} exists \nsuch that $u -e^{t\\Delta} u_0 \\in \nL^\\infty(0,T ; L^q_{s_c}(\\mathbb{R}^d)) \\cap C((0,T] ; L^q_{s_c}(\\mathbb{R}^d))$ and \n$\\|u\\|_{\\mathcal{K}^s(T)} \\le M.$ \nMoreover, the solution satisfies the following properties:\n\\begin{enumerate}[$(i)$]\n\\item $u -e^{t\\Delta} u_0 \\in L^\\infty(0,T ; L^q_{\\sigma}(\\mathbb{R}^d))$ for $\\sigma$ such that \n\\begin{equation}\t\\label{l:exist.crt.csig}\n\ts_c \\le \\sigma \\le \\alpha s -\\gamma.\n\\end{equation}\n\\item\n$u -e^{t\\Delta} u_0 \\in C([0,T) ; L^q_{\\sigma}(\\mathbb{R}^d))$ \nand $\\Lim_{t\\to0} \\|u(t) -e^{t\\Delta} u_0\\|_{L^q_{\\sigma}} = 0$ for $\\sigma$ such that \n\\eqref{l:exist.crt.csig} and $\\sigma>s_c.$ \n\\item $\\displaystyle \\lim_{t\\to0} u(t) = u_0$ in the sense of distributions. \n\\item Let $\\gamma\\ge0.$ Then the solution $u$ satisfies \n\\[\n\\sup_{0<t<T} \n\tt^{\\frac{s_c(p_{\\theta} ) - s_{\\theta}}{2}} \\|u(t)\\|_{L^{p_{\\theta}}_{s_{\\theta}}} < \\infty\n\\]\nwhere \n\\begin{equation}\\nonumber\n\tp_{\\theta} = \\frac{q}{\\theta}, \\quad \n\t\\theta s \\le s_{\\theta} \\le s \\quad\\text{and}\\quad 0\\le \\theta \\le 1.\n\\end{equation}\nIn particular, if $\\gamma\\ge0,$ $u(t) \\in L^\\infty(\\mathbb{R}^d)$ for $t>0.$\n\\end{enumerate}\n\\end{lem}\n\\begin{rem}\\label{r:exist}\nTo meet \\eqref{l:exist.crt.c0}, it suffices to take $M=2\\rho$ and \n\\[\nM < \\min\\left\\{  (2C_0)^{-\\frac{1}{\\alpha-1}}, \\, \n                        (2C_1)^{- \\frac{1}{\\alpha-1}} \\right\\}.\n\\]\n\\end{rem}\n\\begin{rem}\nIf $u_0\\in L^q_{s_c}(\\mathbb{R}^d)$, then $u_0$ satisfies the assumptions of \nLemma \\ref{l:exist.crt} with $T=\\infty.$ Indeed, letting $q\\equiv q,$ $p\\equiv q,$ \n$s\\equiv s_c$ and $s'\\equiv s$ in Lemma \\ref{l:wLpLq}, we obtain \n\\begin{equation}\\nonumber\n\t\\|e^{t\\Delta} u_0\\|_{L^q_s} \\le C t^{-\\frac{s_c-s}2} \\|u_0\\|_{L^q_{s_c}}\n\\end{equation}\nprovided that $-\\frac{d}{q} < s\\le s_c < d(1-\\frac1{q}),$ i.e., \n$\\gamma>-2$ and $\\alpha>\\alpha_F(d,\\gamma).$ \nThus, $e^{t\\Delta} u_0 \\in \\mathcal{K}^s.$ \n\\end{rem}\n\\begin{proof}[Proof of Lemma \\ref{l:exist.crt}]\nSetting the metric $d(u,v) := \\|u-v\\|_{\\mathcal{K}^s(T)}$, we may show that $(\\mathcal{K}^s(T),d)$ is a nonempty complete metric space. Let\n$X_M := \\{ u \\in\\mathcal{K}^s(T) \\,;\\,  \\|u\\|_{\\mathcal{K}^s(T)} \\le M \\}$ be the closed ball in $\\mathcal{K}^s(T)$ centered at the origin with radius $M$.\nWe prove that the map defined in \\eqref{map} has a fixed point in $X_M.$ \nThanks to Lemma \\ref{l:Kato.est} and \\eqref{l:exist.crt.c0}, we have \n\\begin{equation}\\nonumbe\n\\begin{aligned}\n\\|\\Phi (u)\\|_{\\mathcal{K}^s(T)} \n\\le \\|e^{t\\Delta} u_0 \\|_{\\mathcal{K}^s(T)} \n\t+ C_0 \\|u\\|_{\\mathcal{K}^s(T)}^\\alpha \n\\le \\rho + C_0 M^\\alpha \\le M\n\\end{aligned}\\end{equation} \nand \n\\begin{equation}\\label{t:HH.LWP.pr.Lip'}\n\\|\\Phi (u)-\\Phi (v)\\|_{\\mathcal{K}^s(T)} \n\\le C_1 \\left( \\|u\\|_{\\mathcal{K}^s(T)}^{\\alpha-1}\n\t+\\|v\\|_{\\mathcal{K}^s(T)}^{\\alpha-1} \\right) \n\t\\|u-v\\|_{\\mathcal{K}^s(T)}\n\\le 2 C_1 M^{\\alpha-1} \\|u-v\\|_{\\mathcal{K}^s(T)}\n\\end{equation}\nfor any $u, v\\in X_M,$ where $2 C_1 M^{\\alpha-1}<1.$ \nThese prove that $\\Phi(u) \\in X_M$ and that \n$\\Phi$ is a contraction mapping in $X_M.$ \nThus, Banach's fixed point theorem ensures the existence of \na unique fixed point $u$ for the map $\\Phi$ in $X_M,$ \nprovided that $q$ and $s$ satisfy \\eqref{l:Kato.est.c1} and \\eqref{l:Kato.est.c2}. \nThe fixed point $u$ also satisfies, by construction, the estimate \n$\\|u\\|_{\\mathcal{K}^s(T)} \\le M.$ \n\nHaving obtained a fixed point in $\\mathcal{K}^s(T)$ for some $T,$ \nwe have $u -e^{t\\Delta} u_0 \\in L^\\infty(0,T;L^q_{s_c}(\\mathbb{R}^d))$ by Lemma \n\\ref{l:crt.est}, provided further that \\eqref{l:crt.est.c1} and \\eqref{l:crt.est.c2} are satisfied. \nWe see that $\\frac1{q} < \\frac{2}{d(\\alpha-1)}$, \n$q>0,$ $\\alpha>1$ and $\\gamma>-2$ imply \n\\[\n\t\\max\\left\\{\\frac{\\gamma}{\\alpha-1}, \\, - \\frac{d}{q} \\right\\} \n\t\t< \\frac{s_c+\\gamma}{\\alpha} = s_c - \\frac{d(\\alpha-1)}{\\alpha} \\left(\\frac{2}{d(\\alpha-1)} - \\frac1{q} \\right) \n\\]\nso $\\frac{s_c+\\gamma}{\\alpha}$ is the stronger lower bound for $s.$ \nThus, $s$ must satisfy \\eqref{t:HH.LWP.c2}. \nCombining \\eqref{l:Kato.est.c1} and \\eqref{l:crt.est.c1}, we end up with \n\\begin{equation}\\label{t:HH.LWP:pr1}\n\t\\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\,\n\t\t\\frac1{\\alpha} \\left(1 - \\frac{\\gamma}{d(\\alpha-1)}\\right), \\, \n\t\t\\frac1{\\alpha-1} \\left(1-\\frac{2+\\gamma}{d(\\alpha-1)} \\right)\\right\\}, \n\\end{equation}\nwhich in fact amounts to \\eqref{t:HH.LWP.c1}. \n\t\\smallbreak\nWe next prove the assertion $(i)$--$(iii).$ Fix a solution $u \\in \\mathcal{K}^s(T)$ \nwith $q$ and $s$ as in \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}. We have \n\\begin{equation}\\label{t:HH.LWP:pr2}\n\\begin{aligned}\n\\|N(u)(t)\\|_{L^q_{\\sigma}} \n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 (\\alpha s -\\gamma- \\sigma)} \n\t \\| u(\\tau) \\|_{L^q_s}^{\\alpha} d\\tau \\\\\n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 (\\alpha s -\\gamma- \\sigma)} \n\t\\tau^{-\\frac{(s_c-s)\\alpha}{2}} d\\tau \\times  \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha} \\\\\n&= C  t^{\\frac{\\sigma-s_c}2} B\\left(\\frac{(\\alpha-1)(s_c-s) + \\sigma-s}2, \\, 1 - \\frac{(s_c-s)\\alpha}2 \\right)  \n\t\\times  \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha}, \n\\end{aligned}\n\\end{equation}\nthanks to Lemma \\ref{l:wLpLq} with $q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s' \\equiv \\sigma$ and $s\\equiv \\alpha s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < \\sigma \\le \\alpha s -\\gamma < d(1-\\frac{\\alpha}{q}).$ \nThe power of $t$ in the final line is non-negative if $\\sigma \\ge s_c.$ \nThe use of Lemma \\ref{l:wLpLq} along with the convergence of the beta function require, \nin addition to \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}, that $\\sigma$ satisfies\n\\eqref{l:exist.crt.csig}. For such a $\\sigma$ to exist, one needs \n$\\frac{s_c+\\gamma}{\\alpha}\\le s,$ which is assured by \\eqref{t:HH.LWP.c2}. \nIf $\\sigma>s_c,$ (i.e., if $\\frac{s_c+\\gamma}{\\alpha}< s,$) then the power of $t$ is positive, thus the right-hand side of \n\\eqref{t:HH.LWP:pr2} goes to zero as $t\\to0.$ Hence, the assertions $(ii)$ \nand $(iii)$ are proved. \n\t\\smallbreak\nFinally, we prove the assertion $(iv).$ Fix a solution $u \\in \\mathcal{K}^s(T)$ with \n$q$ and $s$ as in \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}. \nHere, we notice that under $\\gamma>0,$ the lower bound of \\eqref{t:HH.LWP.c2} \nalways satisfies \n\\begin{equation}\\nonumber\n\t\\max\\left\\{0, \\, \\frac{\\gamma}{\\alpha}\\right\\} \n\t\\le s_c - \\frac{d(\\alpha-1)}{\\alpha} \\left(\\frac{2}{d(\\alpha-1)} - \\frac1{q} \\right),\n\\end{equation}\nwhich implies that the condition \n\\eqref{l:b.strap:p=inf} of Lemma \\ref{l:b.strap} is always satisfied as well. Thus, \nLemma \\ref{l:b.strap} immediately implies \n\\begin{equation}\\nonumber\n\t\\sup_{t\\in [0,T)} t^{\\frac{s_c(\\infty)-s'}{2} } \\|u(t)\\|_{L^{\\infty}_{s'}} < \\infty\n\\end{equation}\nfor \n\\begin{equation}\\nonumber\n\t0\\le s' \\le \\min\\{s, \\, \\alpha s -\\gamma\\}.\n\\end{equation}\nWe also have $u\\in \\mathcal{K}^s(T)$ by assumption. Thus, the conclusion follows from \nProposition \\ref{p:wL.sp} $(3).$ \n\\end{proof}\nWe start by proving the uniqueness of our solution. \n\t\\subsubsection{Proof of $(ii)$}\nLet $T>0$ be given and fixed. We prove the uniqueness in $\\mathcal{K}^s(T).$ Under conditions \\eqref{t:HH.LWP.c0}, \n\\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}, \nlet $u$ and $v$ be two solutions to \\eqref{integral-eq} \nbelonging to $C([0,T] ; L^q_{s_c}(\\mathbb{R}^d)) \\cap \\mathcal{K}^s(T)$ \nwith the same initial data $u_0 \\in L^q_{s_c}(\\mathbb{R}^d)$ \n($u_0 \\in \\mathcal{L}^{\\infty}_{s_c}(\\mathbb{R}^d)$ if $q=\\infty$)\n\\footnote{We assume $u_0 \\in \\mathcal{L}^{\\infty}_{s_c}(\\mathbb{R}^d)$ if $q=\\infty$ in \norder to utilize the density, which is needed in the proof of \\eqref{id.lmt.K} and \\eqref{id.lmt.crt}.}\nsuch that \n\\begin{equation}\\nonumber\n\t\\|u\\|_{\\mathcal{K}^s(T)}+\\|v\\|_{\\mathcal{K}^s(T)} \\le K,\n\\end{equation}\nfor some positive constant $K.$ \nLet us recall that we have the following two limits at our disposal:\n\\begin{equation}\\label{id.lmt.K}\n\t\\lim_{T\\to0} \\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)} = 0\n\\end{equation}\nand \n\\begin{equation}\\label{id.lmt.crt}\n\t\\lim_{T\\to0} \\|u - e^{t\\Delta} u_0\\|_{L^\\infty(0,T;L^q_{s_c})} = 0. \n\\end{equation}\nThe former is the well-known fact stemming from the density of \n$C_0^\\infty(\\mathbb{R}^d)$ in $L^q_{s_c}(\\mathbb{R}^d)$ (See Proposition \\ref{p:wL.sp} in Appendix). \nThe latter is shown by the triangle inequality and the continuity at $t=0$ of solutions \nfor both the linear and nonlinear problems. \nLet $w :=u-v.$ By \\eqref{diff.pt.est}, we have \n\\begin{equation}\\nonumber\n\t|F(u)-F(v)| \n\t\\le C |e^{t\\Delta} u_0|^{\\alpha-1} |u-v| \n\t+ C (|u-e^{t\\Delta} u_0|^{\\alpha-1} + |v-e^{t\\Delta} u_0|^{\\alpha-1})|u-v|,\n\\end{equation}\nwhich implies that $|w| \\le C( I_1 + I_2 + I_3)$ (thanks to the maximum principle), where  \n\\begin{equation}\\nonumber\n\\begin{aligned}\n&\tI_1 := \\int_0^{t} e^{(t-\\tau)\\Delta} \n\t\t\\left\\{|\\cdot|^{\\gamma} |e^{t\\Delta} u_0|^{\\alpha-1} |w| \\right\\} \\, d\\tau, \\\\\n&\tI_2 :=  \\int_0^{t} e^{(t-\\tau)\\Delta} \n\t\t\\left\\{|\\cdot|^{\\gamma} |u-e^{t\\Delta} u_0|^{\\alpha-1} |w| \\right\\} \\, d\\tau \\\\\n\t\t\\text{and}\\quad\n&\tI_3 :=  \\int_0^{t} e^{(t-\\tau)\\Delta} \n\t\t\\left\\{|\\cdot|^{\\gamma} |v-e^{t\\Delta} u_0|^{\\alpha-1} |w| \\right\\} \\, d\\tau. \n\\end{aligned}\n\\end{equation}\nGiven $q$ and $s$ satisfying \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}, \nwe may always choose $\\theta$ so that \\eqref{l:Kato.est.c1'} and \\eqref{l:Kato.est.c2'} \nare satisfied. Indeed, \\eqref{l:Kato.est.c1'} and \\eqref{l:Kato.est.c2'} become \n\\eqref{l:Kato.est.c1} and \\eqref{l:Kato.est.c2} as $\\theta\\to1,$ respectively, \nwhich are weaker than the assumptions on $q$ and $s$ in Theorem \\ref{t:HH.LWP}. \nThe only condition that has to be considered independently is \n$s_c - \\frac{d}{\\theta} \\left( \\frac{2}{d(\\alpha-1)} -\\frac{1}{q} \\right)  \\le s$ in \n\\eqref{l:Kato.est.c2'} (as this is not a strict inequality), \nbut this causes no problem since \n$s_c - \\frac{d}{\\theta} \\left( \\frac{2}{d(\\alpha-1)} -\\frac{1}{q} \\right) \n\\le s_c - \\frac{d(\\alpha-1)}{\\alpha} \\left(\\frac{2}{d(\\alpha-1)} - \\frac1{q} \\right)$\nholds for any $\\theta \\in (0,1].$ \nThus, we may use estimate \\eqref{l:Kato.est2} freely for our $q$ and $s.$ \n\nBy the same calculation leading to \\eqref{l:Kato.est1}, we deduce that \n\\begin{equation}\\label{pr.uni.1}\n\t\\|I_1\\|_{\\mathcal{K}^s(T)} \n\t\\le C \\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}^{\\alpha-1} \\|w\\|_{\\mathcal{K}^s(T)}. \n\\end{equation}\nFor $I_2,$ estimate \\eqref{l:Kato.est2} implies \n\\begin{equation}\\label{pr.uni.2}\n\\begin{aligned}\n\\|I_2\\|_{\\mathcal{K}^s(T)} \n&\\le C \\|u-e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}^{\\theta(\\alpha-1)} \n\t\t\\|u-e^{t\\Delta} u_0\\|_{L^\\infty(0,T; L^q_{s_c}) }^{(1-\\theta)(\\alpha-1)} \n\t \\|w\\|_{\\mathcal{K}^s(T)}  \\\\\n&\\le C K^{\\theta(\\alpha-1)} \n\t\\|u-e^{t\\Delta} u_0\\|_{L^\\infty(0,T; L^q_{s_c}) }^{(1-\\theta)(\\alpha-1)} \n\t\\|w\\|_{\\mathcal{K}^s(T)}. \n\\end{aligned}\n\\end{equation}\nSimilarly, we have \n\\begin{equation}\\label{pr.uni.3}\n\\begin{aligned}\n\\|I_3\\|_{\\mathcal{K}^s(T)} \n\\le C K^{\\theta(\\alpha-1)} \n\t\\, \\|v-e^{t\\Delta} u_0\\|_{L^\\infty(0,T; L^q_{s_c}) }^{(1-\\theta)(\\alpha-1)} \n\t\\, \\|w\\|_{\\mathcal{K}^s(T)}. \n\\end{aligned}\n\\end{equation}\nGathering \\eqref{pr.uni.1}, \\eqref{pr.uni.2} and \\eqref{pr.uni.3}, we deduce that \nthere exists some positive constant $C$ \nindependent of $T,$ $u_0,$ $u$ and $v$ such that \n\\begin{equation}\\nonumber\n\\|w\\|_{\\mathcal{K}^s(T)} \n\\le C \\mathcal{N}(T, u_0, u, v)\n\t\\|w\\|_{\\mathcal{K}^s(T)} \n\\end{equation}\nwhere\n\\begin{equation}\\nonumber\n\\mathcal{N}(T, u_0, u, v) := \n \\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}^{\\alpha-1} \n\t+\\|u-e^{t\\Delta} u_0\\|_{L^\\infty(0,T; L^q_{s_c}) }^{(1-\\theta)(\\alpha-1)} \n\t+ \\|v-e^{t\\Delta} u_0\\|_{L^\\infty(0,T; L^q_{s_c}) }^{(1-\\theta)(\\alpha-1)}.  \n\\end{equation}\nSince $0<\\theta<1,$ the above quantity goes to zero as \n$T$ tends to zero, thanks to \\eqref{id.lmt.crt} and \\eqref{id.lmt.K}. \nThus, there exists some $T'$ such that \n\\begin{equation}\\nonumber\n\\|w\\|_{\\mathcal{K}^s(T')} \n\t\\le \\frac1{2} \\|w\\|_{\\mathcal{K}^s(T')}\n\\end{equation}\nfor instance, which implies the uniqueness on the interval $[0,T'].$ Set \n\\begin{equation}\\nonumber\nT^* = \\sup \\{t\\in [0,T] \\,; \\, u(\\tau) = v(\\tau) , \\ 0\\le \\tau \\le t\\}.\n\\end{equation}\nThe preceding argument shows that $T^*>0.$\nNow assume by contradiction that $T^*<T.$ By continuity of $u$ and $v,$ \nwe have $u(T^*) = v(T^*).$ \nSetting $u^*(t) = u(t+T^*)$ and $v^*(t) = v(t+T^*),$ we may express the solutions as \n\\begin{align*}\\nonumber\n&\tu^*(t) = e^{t\\Delta} u(T^*) + \\int_0^t e^{(t-\\tau)\\Delta} \n\t\\left\\{ |\\cdot|^{\\gamma} F(u(T^*+\\tau,\\cdot)) \\right\\} d\\tau \\\\\n\t\t\\text{and} \\quad\n&\tv^*(t) = e^{t\\Delta} u(T^*) + \\int_0^t e^{(t-\\tau)\\Delta} \n\t\\left\\{ |\\cdot|^{\\gamma} F(v(T^*+\\tau,\\cdot)) \\right\\} d\\tau,  \n\\end{align*}\nwhere $0\\le t < T - T^*.$ \nBy a similar calculation as above, we may show that \n\\begin{equation}\\nonumber\n\\|u^* - v^*\\|_{\\mathcal{K}^s(T)} \n\t\\le \\mathcal{N}(T, u(T^*), u, v)  \n\t\t\\|u^* - v^*\\|_{\\mathcal{K}^s(T)}, \n\\end{equation}\nwhich implies again that there exist some $T'$ such that $u^*(t) = v^*(t)$ \nfor $t\\in [0,T'],$ i.e., $u(t) = v(t)$ for $t \\in (T^*, T^* +T'),$ a contradiction. \nThus, $u(t)=v(t)$ on the whole interval $[0,T].$ \nThis completes the proof of Theorem \\ref{t:HH.LWP} $(ii).$  \n\n\t\\subsubsection{Proof of $(i)$} \nLet $u_0 \\in L^q_{s_c}(\\mathbb{R}^d)$ \n($u_0 \\in \\mathcal{L}^{\\infty}_{s_c}(\\mathbb{R}^d)$ if $q=\\infty$). \nWe recall that $C_0^\\infty(\\mathbb{R}^d)$ is dense in the space $L^q_{s_c}(\\mathbb{R}^d)$ \nby Proposition \\ref{p:wL.sp}, which ensures the property \\eqref{id.lmt.K}. \nThus, there exists some real number $T$ that is \nsmall enough so that $\\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}\\le \\rho.$ \nNow Lemma \\ref{l:exist.crt} asserts that \n\\begin{equation}\\nonumber\n\\|u\\|_{L^\\infty(0,T \\,;\\, L^q_{s_c})} \n\t\\le \\|e^{t\\Delta} u_0\\|_{L^\\infty(0,T \\,;\\, L^q_{s_c})} + C_2 \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha}\n\t\\le \\|u_0\\|_{L^q_{s_c}} + C_2 M^{\\alpha}. \n\\end{equation}\nThe time-continuity at $t=0$ follows from a well-known argument \n(see \\cites{OkaTsu2016, Tsu2011} for example). \nThus, $u$ is an $L^q_{s_c}(\\mathbb{R}^d)$-mild solution to \\eqref{HH} on $[0,T]$ \nsuch that $\\|u\\|_{\\mathcal{K}^s(T)}\\le M.$ \nTo deduce the estimate \\eqref{t:HH.LWP.est}, \nit suffices to take $\\rho=\\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}$ \nand $M$ as in Remark \\ref{r:exist}. \nGiven $u_0 \\in L^q_{s_c}(\\mathbb{R}^d),$ let the maximal existence time \n$T_m = T_m (u_0)$ be defined by \\eqref{d:Tm} with $\\tilde s = s_c.$ \nBy a standard argument, uniqueness ensures that \nthe solution can be extended to the maximal interval $[0,T_m).$ \n\n\t\\subsubsection{Proof of $(iii)$}\nGiven two initial data $u_0, v_0 \\in L^q_{s_c}(\\mathbb{R}^d),$ \nwe next show the Lipschitz continuity of the flow map. \nLet $u$ and $v$ be two solutions associated with the initial data $u_0$ and $v_0,$ \nrespectively, constructed in (i) with the estimate \n$\\|u\\|_{\\mathcal{K}^s(T)}\\le 2\\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}.$ \nLet $w := u-v$ and $w_0 := u_0-v_0.$ \nWe carry out the same calculations as before to \nsee that there exists a positive constant $C_3$ such that \n\\begin{align*}\n\\|w\\|_{L^\\infty(0,T; L^q_{s_c})\\cap \\mathcal{K}^s(T)} \n&\\le \\|e^{t\\Delta}w_0\\|_{L^\\infty(0,T; L^q_{s_c})\\cap \\mathcal{K}^s(T)} \n\t+ C_3 \\left( \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha-1}\n\t\t+\\|v \\|_{\\mathcal{K}^s(T)}^{\\alpha-1}\\right) \\|w\\|_{\\mathcal{K}^s(T)} \\\\\n&\\le \\|w_0\\|_{L^q_{s_c}} \n\t+ 2 C_3 M^{\\alpha-1} \\|w\\|_{\\mathcal{K}^s(T)}, \n\\end{align*}\nwhere $M= \\max \\{ \\|e^{t\\Delta} u_0 \\|_{\\mathcal{K}^s(T)}, \n\t\\, \\|e^{t\\Delta} v_0 \\|_{\\mathcal{K}^s(T)}\\}.$ \nBy taking $T$ smaller if necessary \n($2 C_3 M^{\\alpha-1}\\le \\frac12$ for instance), we deduce \nthe Lipschitz stability on the short time-interval $[0,T].$ \n\n\t\\subsubsection{Proof of $(iv)$}\nWe prove the blow-up criterion by a contradiction argument. \nLet $T_m<\\infty$ and suppose that $\\|u\\|_{\\mathcal{K}^s(T_m)} <\\infty$ holds.  \nLet $u$ be a maximal solution and let $t_0 \\in (0,T_m),$ to be fixed later.  \nWe aim to prove \nthere exists an $\\varepsilon>0$ such that \n\\begin{equation}\\label{buc-aim}\n\t\\|e^{t\\Delta} u(t_0)\\|_{\\mathcal{K}^s(T_m-t_0+\\varepsilon)} \\le \\rho,\n\\end{equation}\nwhere $\\rho > 0$ is the constant as in Lemma \\ref{l:exist.crt}. \nOnce \\eqref{buc-aim} is proved, \nthe solution $u$ can be smoothly extended to $T_m+\\varepsilon.$ \nMoreover, $u$ is unique in \n$C([0,T_m+\\varepsilon] ; L^q_{s_c}(\\mathbb{R}^d)) \\cap \\mathcal{K}^s(T_m+\\varepsilon)$ by $(ii),$ \nwhich contradicts the definition of $T_m.$ \nThus, $\\|u\\|_{\\mathcal{K}^s(T_m)}=\\infty$ if $T_m<\\infty.$ \n\nLet us concentrate on proving \\eqref{buc-aim}. We may express the maximal solution as follows: \n\\begin{equation}\\nonumber\n\tu(t+t_0) = e^{t\\Delta} u(t_0) + \\int_0^t e^{(t-\\tau)\\Delta} \\left\\{ |\\cdot|^{\\gamma} F(u(t_0+\\tau)) \\right\\} d\\tau, \\quad 0\\le t < T_m - t_0.\n\\end{equation}\nThus, we have \n\\begin{align*}\\nonumber\n\t\\|e^{t\\Delta}u(t_0)&\\|_{\\mathcal{K}^s(T_m-t_0)}  \\\\\n\t&\\le \\|u(\\cdot+t_0)\\|_{\\mathcal{K}^s(T_m-t_0)} \n\t+ \\left\\| \\int_0^t e^{(t-\\tau)\\Delta} \\left\\{ |\\cdot|^{\\gamma} F(u(t_0+\\tau)) \\right\\} d\\tau \\right\\|_{\\mathcal{K}^s(T_m-t_0)}. \n\\end{align*}\nFor the first term, we have \n\\begin{equation}\\label{pr.iv.1}\n\\begin{aligned}\n\t\\|u(\\cdot+t_0)&\\|_{\\mathcal{K}^s(T_m-t_0)} \n\t= \\sup_{0\\le t \\le T_m - t_0} t^{\\frac{s_c-s}2} \\|u(t + t_0)\\|_{L^q}\n\t= \\sup_{t_0\\le s\\le T_m} (s-t_0)^{\\frac{s_c-s}2} \\|u(s)\\|_{L^q} \\\\\n\t&\\le \\left(\\frac{T_m-t_0}{t_0} \\right)^{\\frac{s_c-s}2} \n\t\t\\sup_{t_0 \\le s\\le T_m} s^{\\frac{s_c-s}2} \\|u(s)\\|_{L^q} \n\t\\le \\left(\\frac{T_m-t_0}{t_0} \\right)^{\\frac{s_c-s}2} \\|u\\|_{\\mathcal{K}^s(T_m)}.\n\\end{aligned}\n\\end{equation}\nFor the second term, Lemma \\ref{l:Kato.est} yields \n\\begin{equation}\\label{pr.iv.2}\n\t\\left\\| \\int_0^t e^{(t-\\tau)\\Delta} \\left\\{ |\\cdot|^{\\gamma} \n\t\t\t\tF(u(t_0+\\tau)) \\right\\} d\\tau \\right\\|_{\\mathcal{K}^s(T_m-t_0)}\n\t\\le C_0 \\|u(\\cdot+t_0)\\|_{\\mathcal{K}^s(T_m-t_0)}^\\alpha.\n\\end{equation}\nSince the right-hand sides in \\eqref{pr.iv.1} and \\eqref{pr.iv.2} go to $0$ as $t_0 \\to T_m$,\nwe may fix some $t_0$ \nclose enough to $T_m$ so that \n\\begin{equation}\\nonumber\n\t\\|e^{t\\Delta} u(t_0)\\|_{\\mathcal{K}^s(T_m-t_0)} \n\t\\le 2^{-\\frac{s_c-s}2}  \\frac{\\rho}{2}.\n\\end{equation}\nLet $\\varepsilon\\in (0, T_m -t_0)$, to be fixed later. \nThen, we have \n\\begin{equation}\\label{pr.iv.4}\n\\begin{aligned}\n\\sup_{2\\varepsilon\\le t\\le T_m - t_0 +\\varepsilon} t^{\\frac{s_c-s}2} \\|e^{t\\Delta} u(t_0)\\|_{L^q_s} \n\t&=  \\sup_{\\varepsilon \\le s\\le T_m - t_0} \\left( \\frac{s+\\varepsilon}{s}\\right)^{\\frac{s_c-s}2} \n\t\t s^{\\frac{s_c-s}2} \\|e^{(s+\\varepsilon)\\Delta} u(t_0)\\|_{L^q_s} \\\\\n\t&\\le \\sup_{\\varepsilon \\le s\\le T_m - t_0} \\left( \\frac{s+\\varepsilon}{s}\\right)^{\\frac{s_c-s}2} \n\t\t\\|e^{t\\Delta} u(t_0)\\|_{\\mathcal{K}^s(T_m-t_0)}\\\\\n\t&\\le 2^{\\frac{s_c-s}2} \n\t\t\\|e^{t\\Delta} u(t_0)\\|_{\\mathcal{K}^s(T_m-t_0)} \\le  \\frac{\\rho}{2},\n \\end{aligned}\n \\end{equation}\nwhere we have used \n$$\\displaystyle{\\sup_{\\varepsilon \\le s\\le T_m - t_0} \\frac{s+\\varepsilon}{s} \\le 2.}$$\nOn the other hand, since $u(t_0) \\in L^q_{s_c}(\\mathbb{R}^d),$ we may fix some $\\varepsilon>0$ \nsuch that \n\\begin{equation}\\label{pr.iv.0}\n\t\\|e^{t\\Delta}u(t_0)\\|_{\\mathcal{K}^s(2\\varepsilon)} \\le \\frac{\\rho}2, \n\\end{equation}\nBy \\eqref{pr.iv.0} and \\eqref{pr.iv.4}, we deduce that \n\\[\n\\begin{split}\n\t\\|e^{t\\Delta} u(t_0)\\|_{\\mathcal{K}^s(T_m-t_0+\\varepsilon)} \n\t& \\le \\|e^{t\\Delta}u(t_0)\\|_{\\mathcal{K}^s(2\\varepsilon)} + \\sup_{2\\varepsilon\\le t\\le T_m - t_0 +\\varepsilon} t^{\\frac{s_c-s}2} \\|e^{t\\Delta} u(t_0)\\|_{L^q_s} \\\\\n\t& \\le \\frac{\\rho}2 + \\frac{\\rho}2 = \\rho,\n\\end{split}\n\\]\nwhich proves \\eqref{buc-aim}. \n\n\t\\subsubsection{Proof of $(v)$}\nTaking $T=\\infty$ in Lemma \\ref{l:exist.crt}, we deduce the global existence. \nLastly, we show that if $T_m=\\infty,$ then the solution is dissipative. \nWe sketch the proof, as most of the computations are similar to the previous ones. \nWe take $\\{u_{0n}\\}_{n\\ge0} \\subset C_0^\\infty(\\mathbb{R}^d)$ such that \n$u_{0n} \\to u_0$ in $L^q_{s_c}(\\mathbb{R}^d)$ and decompose the integral equation into \n\\begin{equation}\\nonumber\\begin{aligned}\nu(t) =  e^{t\\Delta} u_{0n} + e^{t\\Delta} (u_0-u_{0n}) \n&+ e^{(t-t')\\Delta} \\int_{0}^{t'} e^{(t'-\\tau)\\Delta} \\left( |\\cdot|^{\\gamma} F(u(\\tau)) \\right)d\\tau \\\\\n&+ \\int_{t'}^{t} e^{(t-\\tau)\\Delta} \\left( |\\cdot|^{\\gamma} F(u(\\tau)) \\right)d\\tau,\n\\end{aligned}\\end{equation}\nwhere $0 < t' < t.$ \nThe first and second linear terms obviously tend to 0 as $n\\to \\infty$ and $t\\to\\infty.$\nOn the other hand, we may let $t'$ so close to $t$ so that the fourth term is small. \nNow that $t'$ is fixed, the third term can be written as $e^{(t-t')\\Delta} f(t')$ with \n$f(t') \\in L^q_{s_c}(\\mathbb{R}^d)$, \nso we may use the semigroup property of $e^{t\\Delta}$ and an \napproximation argument again. \nThis completes the proof of the theorem. \n\n\n\n\\subsection{Proof of Theorem \\ref{t:HH.LWP.sub}}\n\\begin{lem}\\label{l:exist.sub}\nLet real numbers $T\\in (0,\\infty),$ $\\rho>0$ and $M>0$ satisfy \n\\begin{equation}\\label{l:exist.sub.c0}\n\t\\rho + \\tilde C_0 T^{\\frac{\\alpha-1}2(s_c-\\tilde s)} M^\\alpha \\le M \n\t\t\\quad\\text{and}\\quad\n\t2 \\tilde C_1 T^{\\frac{\\alpha-1}2(s_c-\\tilde s)} M^{\\alpha-1} <1,\n\\end{equation}\nwhere $\\tilde C_0$ and $\\tilde C_1$ are as in Lemma \\ref{l:Kato.est.sub}. \nUnder conditions \\eqref{t:HH.LWP.c0}, \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}, \nlet $u_0 \\in \\mathcal{S}'(\\mathbb{R}^d)$ be such that \n$e^{t\\Delta}u_0 \\in \\tilde{\\mathcal{K}}^s(T)$ for $T$ fixed as above.  \nIf $\\|e^{t\\Delta}u_0\\|_{\\tilde{\\mathcal{K}}^s(T)}\\le \\rho,$ \nthen a solution $u$ to \\eqref{HH} exists \nsuch that $u -e^{t\\Delta} u_0 \\in C([0,T] ; L^q_{\\tilde s}(\\mathbb{R}^d))$ and \n$\\|u\\|_{\\tilde{\\mathcal{K}}^s(T)} \\le M.$ \n\\end{lem}\n\\begin{rem}\\label{r:exist.sub}\nTo meet condition \\eqref{l:exist.sub.c0}, it suffices to take $M=2\\rho$ and \n$T$ such that \n\\[\nT < \\min\\left\\{  (2^{\\alpha}\\tilde C_0)^{-\\frac{2}{(\\alpha-1)(s_c-\\tilde s)}} , \n\t\t\t\\, (2^{\\alpha}\\tilde C_1)^{-\\frac{2}{(\\alpha-1)(s_c-\\tilde s)}} \n\t\t\t\\right\\} \\rho^{-\\frac2{s_c-\\tilde s}}.\n\\]\n\\end{rem}\n\\begin{proof}[Proof of Lemma \\ref{l:exist.sub}]\nSetting the metric $d(u,v) := \\|u-v\\|_{\\tilde{\\mathcal{K}}^s(T)}$, we may show that $(\\tilde{\\mathcal{K}}^s(T),d)$ is a nonempty complete metric space. Let\n$X_M := \\{ u \\in\\mathcal{K}^s(T) \\,;\\,  \\|u\\|_{\\tilde{\\mathcal{K}}^s(T)} \\le M \\}$ be the closed ball in $\\tilde{\\mathcal{K}}^s(T)$ centered at the origin with radius $M$.\nSimilarly to the critical case, we may prove \nthat the map defined in \\eqref{map} has a fixed point in $\\tilde X_M,$ \nthanks to Lemma \\ref{l:Kato.est.sub} and \\eqref{l:exist.sub.c0}. \nThus, Banach's fixed point theorem ensures the existence of \na unique fixed point $u$ for the map $\\Phi$ in $\\tilde X_M.$ \n\nHaving obtained a fixed point in $\\tilde{\\mathcal{K}}^s(T),$ \nwe deduce $u -e^{t\\Delta} u_0 \\in L^\\infty(0,T;L^q_{\\tilde s}(\\mathbb{R}^d))$ \nthanks to Lemma \\ref{l:subcrt.est}, provided further that \n\\eqref{t:HH.LWP.sub.cs}, \\eqref{l:subcrt.est.c1} and \\eqref{l:subcrt.est.c2} \nare satisfied. \nWe see that $s<\\tilde s < s_c$ imply \n\\[\n\t\\max\\left\\{\\frac{\\gamma}{\\alpha-1}, \\, \\tilde s - \\frac2{\\alpha} \\right\\} \n\t< \\frac{\\tilde s+\\gamma}{\\alpha} \n\\]\nso $\\frac{s_c+\\gamma}{\\alpha}$ is a new lower bound for $s.$\nIn conjunction with this stronger lower bound $\\frac{s_c+\\gamma}{\\alpha} \\le s,$ \nthere also appears a new upper bound for $\\frac1{q}.$ \nMore precisely, for such an $s$ satisfying \\eqref{l:Kato.est.sub.c2}\nand \\eqref{l:subcrt.est.c2} to exist, $q$ must satisfy, \nin addition to \\eqref{l:Kato.est.sub.c1} and \\eqref{l:subcrt.est.c1}, \n\\begin{equation}\\label{t:HH.LWP.sub:pr2}\n\t\\frac1{q} < \\frac1{d\\alpha} \\left(\\frac{2\\alpha + \\gamma}{\\alpha-1} -\\tilde s\\right). \n\\end{equation}\nIndeed, $\\frac{\\tilde s+\\gamma}{\\alpha} < s_c$ \nis equivalent to $\\frac1{q} <\\frac1{d\\alpha} (\\frac{2\\alpha + \\gamma}{\\alpha-1} -\\tilde s).$ We notice that \n$\\frac1{\\alpha} (1-\\frac{\\tilde s}{d}) < \\frac1{\\alpha} (1-\\frac{\\gamma}{d(\\alpha-1)} )$ \nand \n$\\frac1{d\\alpha} (\\frac{2\\alpha + \\gamma}{\\alpha-1} -\\tilde s) > \n\\frac1{d} (\\frac{2 + \\gamma}{\\alpha-1} -\\tilde s)$ as $\\frac{\\gamma}{\\alpha-1} < \\tilde s.$ \nThus, combining \\eqref{l:Kato.est.sub.c1} and \\eqref{t:HH.LWP.sub:pr2}, we deduce \nthat the conditions for $q$ are \\eqref{t:HH.LWP.sub.c1}\n\\end{proof}\n\nWe omit the proofs of $(i),$ $(ii)$ and $(iii)$ of Theorem \\ref{t:HH.LWP.sub} \nas they are standard. We only prove $(iv).$\n\n\t\\subsubsection{Proof of $(iv)$}\nLet $u_0 \\in L^q_{\\tilde s}(\\mathbb{R}^d)$ be such that $T_m = T_m(u_0)$ is finite and let \n$u \\in C([0,T_m) ; L^q_{\\tilde s}(\\mathbb{R}^d))$ be the maximal solution of \\eqref{HH}. \nFix $t_0 \\in (0,T_m)$ and so that we may express the maximal solution by \n\\begin{equation}\\nonumber\n\tu(t+t_0) = e^{t\\Delta} u(t_0) + \\int_0^t e^{(t-\\tau)\\Delta} \\left\\{ |\\cdot|^{\\gamma} F(u(t_0+\\tau,\\cdot)) \\right\\} d\\tau, \\quad 0\\le t < T_m - t_0.\n\\end{equation}\nWe observe that \n\\begin{equation}\\nonumber\n\t\\|u(t_0)\\|_{L^q_{\\tilde s}} \n\t\t+ \\tilde C_0 (T_m-t_0)^{\\frac{\\alpha-1}2(s_c-\\tilde s)} M^\\alpha > M\n\\end{equation}\nholds for all $M>0,$ where $\\tilde C_0$ is as in \\eqref{l:Kato.est.sub1}.\nOtherwise there exists $M>0$ such that \n\\begin{equation}\\nonumber\n\t\\|u(t_0)\\|_{L^q_{\\tilde s}} \n\t\t+ \\tilde C_0 (T_m-t_0)^{\\frac{\\alpha-1}2(s_c-\\tilde s)} M^\\alpha \\le M\n\\end{equation}\nso that one may argue as in the proof of existence to obtain a local solution such that \n$\\|u(t + t_0)\\|_{L^q_{\\tilde s}} \\le M$ for $t\\in [0,T_m -t_0]$ and in particular, \n$u(T_m)$ is well-defined in $L^q_s(\\mathbb{R}^d),$ contradicting the definition of $T_m.$ \nLet $M = 2\\|u(t_0)\\|_{L^q_{\\tilde s}}$ so that \n\\begin{equation}\\nonumber\n\t\\|u(t_0)\\|_{L^q_{\\tilde s}} + 2^{\\alpha} \\tilde C_0 \\|u(t_0)\\|_{L^q_{\\tilde s}}^\\alpha (T_m-t_0)^{\\frac{\\alpha-1}2(s_c-\\tilde s)}  > 2 \\|u(t_0)\\|_{L^q_{\\tilde s}}, \n\\end{equation}\nwhich yields \\eqref{t:HH.LWP:Tm}. \nIn particular, $\\|u(t)\\|_{L^q_{\\tilde s}} \\to \\infty$ as $t\\to T_m.$ \nThus, we conclude Theorem \\ref{t:HH.LWP.sub}. \n\n\\subsection{Proof of Theorem \\ref{t:HH.self.sim}}\nLet $\\psi (x) := |x|^{-\\frac{2+\\gamma}{\\alpha-1}}$ for $x\\ne 0$. \nWe first claim that a initial data $u_0$ given by $u_0(x):= c\\psi(x)$ with a sufficiently small $c$ satisfies the all assumptions of $(v)$ in Theorem \\ref{t:HH.LWP} with $T=\\infty,$ thereby generating a global solution to the Cauchy problem (\\ref{HH}) with the initial data $u_0$. Since $\\psi \\in L^1_{loc}(\\mathbb{R}^d)$ as $\\alpha>\\alpha_F(d,\\gamma),$ $\\psi \\in \\mathcal{S}'(\\mathbb{R}^d)$ and $e^{t\\Delta} \\psi$ is well-defined. Since $s<s_c,$ there exist $s_1, s_2 \\in\\mathbb{R}$ such that $s<s_1 < s_c <s_2.$ As in the proof of \\cite[Theorem 1.3]{BenTayWei2017}, we can prove that $\\psi$ can be decomposed into $\\psi = \\psi_1 + \\psi_2,$ \n$\\psi_1 := \\chi_{|x|>1} \\psi$ and $\\psi_2 := \\chi_{|x|<1} \\psi$ \nso that $\\psi_1 \\in L^q_{s_1}(\\mathbb{R}^d)$ and $\\psi_2 \\in L^q_{s_2}(\\mathbb{R}^d).$ \nThis implies that the estimate $\\|e^{\\Delta} \\psi\\|_{L^q_s} \n\t\\le C( \\| \\psi_1\\|_{L^q_{s_1}} + \\| \\psi_2\\|_{L^q_{s_2}})$ holds, \nthanks to Lemma \\ref{l:wLpLq}. By the homogeneity of the data, we deduce\n$\\|e^{t\\Delta} \\psi\\|_{\\mathcal{K}^s} < \\infty.$ Thus, if the constant $c$ is taken small enough so that $(v)$ in Theorem \\ref{t:HH.LWP} is satisfied, the initial data $u_0=c\\psi$ generates a unique global solution to (\\ref{HH}). \n\nLet $\\varphi := \\omega \\psi$ be as in the assumption of Theorem \\ref{t:HH.self.sim}. \nThen we note that $\\varphi$ is homogeneous of degree $-\\frac{2+\\gamma}{\\alpha-1}.$ \nWe show that the global solution $u$ to (\\ref{HH}) with the initial data \n$\\varphi$, which is obtained by $(v)$ in Theorem \\ref{t:HH.LWP}, is also self-similar. To this end, for $\\lambda>0$, \nlet $\\varphi_{\\lambda}$ be defined by \n$\\varphi_{\\lambda} (x) := \\lambda^{\\frac{2-\\gamma}{\\alpha-1}} \\varphi(\\lambda x).$  \nSince the identity $\\|\\varphi_{\\lambda}\\|_{\\mathcal{K}^s} = \\|\\varphi \\|_{\\mathcal{K}^s}$ holds for all $\\lambda>0,$ it follows that $\\varphi_{\\lambda}$ also \nsatisfies the assumptions of $(v)$ in Theorem \\ref{t:HH.LWP}. \nAs $u_\\lambda$ given by \\eqref{scale} is a solution of \\eqref{HH} with initial data \n$\\varphi_{\\lambda},$ and $\\|u_{\\lambda}\\|_{\\mathcal{K}^s} = \\|u \\|_{\\mathcal{K}^s}$ \nfor all $\\lambda>0,$ we deduce that $u$ must be self-similar since $\\varphi_{\\lambda}=\\varphi$. We denote the global self-similar solution $u$ by $u_{\\mathcal{S}}$. The fact $u_{\\mathcal{S}}(t)\\rightarrow\\varphi$ in $\\mathcal{S}'(\\mathbb{R}^d)$ as $t\\rightarrow +0$ follows from $(iii)$ in Lemma \\ref{l:exist.crt}. This completes the proof of Theorem \\ref{t:HH.self.sim}. \n\n\\section{Nonexistence of local positive weak solution}\nIn this section we give a proof of Theorem \\ref{t:nonex}. As the argument is standard, we only give a sketch of the proof. \nFor the details, we refer to \\cite[Proposition 2.4, Theorem 2.5]{II-15}. \n\\subsection{Proof of Theorem \\ref{t:nonex}}\nLet $T\\in (0,1)$. Suppose that the conclusion of Theorem \\ref{t:nonex} does not hold. \nThen there exists a positive weak solution $u$ on $[0,T)$ \n(See Definition \\ref{d:w.sol}). Let \n\\[\n\\psi_T(t,x) := \\eta\\left(\\frac{t}{T}\\right) \\phi\\left(\\frac{x}{\\sqrt{T}}\\right),\n\\]\nwhere $\\eta \\in C^\\infty_0([0,\\infty))$ and $\\phi\\in C^\\infty_0(\\mathbb R^d)$ are such that\n\\[\n\\eta (t)\n:= \n\\begin{cases}\n1,\\quad 0\\le t \\le \\frac12,\\\\\n0,\\quad t\\ge1,\n\\end{cases}\n\\quad \\text{and}\\quad \n\\phi(x)\n:= \n\\begin{cases}\n1,\\quad |x| \\le \\frac12,\\\\\n0,\\quad |x|\\ge1.\n\\end{cases}\n\\]\nLet $l\\in\\mathbb N$ with $l\\ge3$, which will be chosen later. \nWe note that $\\psi_T^l\\in C^{1,2}([0,T)\\times \\mathbb{R}^d)$ and the estimates $|\\partial_t \\{\\psi_T (t,x)\\}^l|\\le \\frac{C}{T} \\psi_T(t,x)^{l-1}$ \nand $|\\partial_{x_j}^2\\{\\psi_T(t,x)^l\\}|\\le \\frac{C}{T} \\psi_T(t,x)^{l-1}$ hold for $j=1,\\ldots, d.$ We define a function $I:[0,T)\\rightarrow \\mathbb{R}_{\\ge 0}$ given by\n\\[\n  I(T):=\\int_{[0,T)\\times \\{|x|<\\sqrt{T}\\}}|x|^{\\gamma} u(t,x)^{\\alpha} \\, \\psi_T^l \\, dtdx.\n\\]\nWe note that $I(T)<\\infty$, since $u\\in L_t^{\\alpha}(0,T;L^{\\alpha}_{\\frac{\\gamma}{\\alpha},loc}(\\mathbb{R}^d))$.\nBy using the weak form (\\ref{weak}) and the above estimates, the estimates hold:\n\\[\n\\begin{aligned}\nI(T) + \\int_{|x|<\\sqrt{T}} u_0(x) \\phi^l\\left(\\frac{x}{\\sqrt{T}}\\right)\\, dx\n& = \n\\left|\\int_{[0,T)\\times \\{|x|<\\sqrt{T}\\}}u(\\partial_t \\psi_T^l + \\Delta \\psi_T^l )\\,dt\\,dx \\right|\\\\\n& \\le \\frac{C}{T}\n\\int_{[0,T)\\times \\{|x|<\\sqrt{T}\\}}|u| \\psi_T^{\\frac{l}{\\alpha}} \\,dt\\,dx.\n\\end{aligned}\n\\]\nHere we choose $l$ as \n\\begin{equation}\\nonumber\n\t-\\frac{l}{\\alpha}+l-2>0, \\quad\\text{i.e.,}\\quad l > \\frac{2\\alpha}{\\alpha-1}.\n\\end{equation}\nBy H\\\"older's inequality and Young's inequality, we may estimate the integral \nin the right-hand side above by \n\\[\n\\begin{aligned}\nT^{-1}&\\int_{[0,T)\\times \\{|x|<\\sqrt{T}\\}}|u| \\psi_T^{\\frac{l}{\\alpha}}\\, dtdx \n\\le I(T)^\\frac{1}{\\alpha} \\cdot T^{-1}K(T)^\\frac{1}{\\alpha'}\n\\le \\frac12 I(T) + \\frac{C}{T^{\\alpha'}}K(T).\n\\end{aligned}\n\\]\nwhere $1= \\frac{1}{\\alpha} + \\frac{1}{\\alpha'}$, i.e., $\\alpha'=\\frac{\\alpha}{\\alpha-1}$, and \n\\[\nK(T) :=  \\int_{[0,T)\\times \\{|x|<\\sqrt{T}\\}}(|x|^{-\\frac{\\gamma}{\\alpha}})^{\\alpha'}\\, dtdx\n=  T\\int_{|x|<\\sqrt{T}}|x|^{-\\frac{\\gamma}{\\alpha-1}}\\, dx=CT^{1-\\frac{\\gamma}{2(\\alpha-1)}+\\frac{d}{2}}\n\\]\ndue to $\\alpha>1+\\gamma\/d$. Summarizing the estimates obtained now, we have\n\\begin{equation}\\label{ineq1}\n\\begin{aligned}\n\\int_{|x|<\\sqrt{T}} u_0(x) \\phi^l\\left(\\frac{x}{\\sqrt{T}}\\right)\\, dx\n \\le \nI(T) + 2\\int_{|x|<\\sqrt{T}} u_0(x) \\phi^l\\left(\\frac{x}{\\sqrt{T}}\\right)\\, dx\n \\le\n C T^{- \\frac{2+\\gamma}{2(\\alpha-1)} + \\frac{d}{2}}.\n\\end{aligned}\n\\end{equation}\nWe now choose the initial data $u_0$ as\n\\[\nu_0(x) := \n\\begin{cases}\n\t|x|^{-\\beta} \\quad & |x|\\le 1,\\\\\n\t0 & \\text{otherwise}\n\\end{cases}\n\\]\nwith \n\\begin{equation}\\label{beta1}\n\t\\beta< \\min\\left\\{s + \\frac{d}{q},d\\right\\}.\n\\end{equation}\nThen $u_0 \\in L^q_s (\\mathbb{R}^d)$ and by $T<1$ and $\\beta<d$, we have\n\\begin{equation}\\label{ineq2}\n\\begin{aligned}\n\t\\int_{|x|<\\sqrt{T}} u_0(x) \\phi^l\\left(\\frac{x}{\\sqrt{T}}\\right)\\, dx\n\t& = T^{-\\frac{\\beta-d}{2}}\t\\int_{|y|<1} |y|^{-\\beta} \\phi^l(y)\\, dx\n\t = C T^{-\\frac{\\beta-d}{2}}.\n\\end{aligned}\n\\end{equation}\nCombining \\eqref{ineq1} and \\eqref{ineq2}, we obtain\n\\begin{equation}\\label{contradiction}\n0< C \\le T^{\\frac{\\beta}{2} - \\frac{2+\\gamma}{2(\\alpha-1)}} \\to 0 \\quad \\text{as }T\\to 0,\n\\end{equation}\nwhere\n\\begin{equation}\\label{beta2}\n\\frac{\\beta}{2} - \\frac{2+\\gamma}{2(\\alpha-1)} >0\\quad \\text{i.e.}\\quad \\beta > \\frac{2+\\gamma}{\\alpha-1},\n\\end{equation}\nwhich leads to a contradiction. Thus the proposition holds if we take \n$\\beta$ satisfying \\eqref{beta1} and \\eqref{beta2}, which amount to $s>s_c$ and $\\alpha>\\alpha_F(d,\\gamma)$. The proof is complete.\n\n\\section{Appendix}\n\\par\nWe list basic properties of the weighted Lebesgue spaces $L^q_s(\\mathbb{R}^d)$. \n\\begin{prop}\n\\label{p:wL.sp}\nLet $s\\in\\mathbb{R}$ and $q\\in [1,\\infty].$ Then the following holds:\n\\begin{enumerate}[$(1)$]\n\\item The space $L^q_{s}(\\mathbb{R}^d)$ is a Banach space. \n\\item $C_0^\\infty(\\mathbb{R}^d)$ is dense in $L^q_{s}(\\mathbb{R}^d)$ if $q$ and $s$ satisfy \n\t\\begin{equation}\\nonumber\n\t\t1\\le q < \\infty\t\\quad\\text{and}\\quad\n\t\t-\\frac{d}{q} < s < d\\left( 1-\\frac{1}{q} \\right).\n\t\\end{equation}\n\\item For $s_1, s_2 \\in \\mathbb{R},$ $q_1, q_2 \\in [1,\\infty],$ we have \n\t\\begin{equation}\\nonumber\n\t\\|f\\|_{L^q_s} \\le \\|f\\|_{L^{q_1}_{s_1}}^{\\theta} \\|f\\|_{L^{q_2}_{s_2}}^{1-\\theta}\n\t\\end{equation}\n\tfor $s = \\theta s_1 + (1-\\theta) s_2,$ \n\t$\\frac1{q} = \\frac{\\theta}{q_1} + \\frac{1-\\theta}{q_2}$ and $\\theta \\in (0,1).$ \n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n$(1)$ The space $L^q_{s}(\\mathbb{R}^d)$ is a Lebesgue space \nwith a measure $d\\mu = |x|^{sq} \\,dx.$ See any standard textbook for the proof of \nits completeness. \\\\\n$(2)$ Recall that the weight $|x|^{sq}$ belongs to \nthe Muckenhoupt class $A_q$ if and only if $- \\frac{d}{q} < s < d(1- \\frac1{q})$ \nwhen $q\\in (1,\\infty),$ and $|x|^{s} \\in A_1$ if and only if $-d<s\\le 0$ when $q=1.$ \nNow the density follows from \\cite{NakTomYab2004}[Theorem 1.1]. \n\\\\\n$(3)$ For $s$ and $q$ as in the assumption, we have \n\\begin{align*}\\nonumber\n\t\\|f\\|_{L^q_s} \n\n\n\t&\\le \\||\\cdot|^{s_1} f\\|_{L^{q_1}}^{\\theta} \\||\\cdot|^{s_2} f\\|_{L^{q_2}}^{1-\\theta} \n\t= \\|f\\|_{L^{q_1}_{s_1}}^{\\theta}  \\|f\\|_{L^{q_2}_{s_2}}^{1-\\theta}. \n\\end{align*}\n\\end{proof}\nThe following pointwise bound is well-known in the literature. \n\\begin{lem}\\label{l:g.unfrm.bnd}\nLet $d\\in\\mathbb N,$ $q\\in[1,\\infty)$ and $a,b,c\\in\\mathbb{R}.$ \nLet $g(x):=(4\\pi)^{-\\frac{d}2} e^{-\\frac{|x|^2}4}.$ \n\\begin{enumerate}[$(1)$]\n\\item \nThere exists a constant $C$ depending only on $d,$ $q,$ $a$ and $b$ such that \n\\begin{equation}\\nonumber\n\t\\sup_{x\\in\\mathbb{R}^d} \\int_{\\mathbb{R}^d} ( |y|^{-a} |x-y|^{b} g(x-y))^q \\,dy \\le C\n\\end{equation}\nprovided that $0\\le a<\\frac{d}{q}$ and $b\\ge0.$ \n\\item \nThere exists a constant $C$ depending only on $d,$ $q$ and $c$ such that \n\\begin{equation}\\nonumber\n\t\\sup_{x\\in\\mathbb{R}^d} \\int_{\\mathbb{R}^d} ( |y|^{-c} g(x-y))^q \\, dy \\le C\n\\end{equation}\nprovided that $0\\le c<\\frac{d}{q}.$\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nIn what follows, we shall use the fact that there exists an absolute constant $C$ such that \n\\begin{equation}\\label{l:g.unfrm.bnd:pr1}\n\tg(x) \\le C \\langle x\\rangle^{-N}\n\\end{equation}\nfor any $N\\in\\mathbb N,$ where $\\langle x\\rangle :=(1+|x|^2)^{\\frac{1}2}.$ Let \n\\begin{equation}\\nonumber\n\t\\begin{aligned}\n\tI(x) &:= \\int_{\\mathbb{R}^d} ( |y|^{-a} |x-y|^{b} g(x-y))^q \\, dy \\\\\n\t&= \\int_{|y|< |x-y|} ( |y|^{-a} |x-y|^{b} g(x-y))^q \\, dy \n\t\t\t+ \\int_{|y|> |x-y|} ( |y|^{-a} |x-y|^{b} g(x-y))^q \\, dy \\\\\n\t&=: I_1 (x) + I_2 (x).\n\t\\end{aligned}\n\\end{equation}\nThanks to \\eqref{l:g.unfrm.bnd:pr1} and $0\\le b,$ we have \n\\begin{align*}\n\tI_1(x) \\le C \\int_{|y|< |x-y|} |y|^{-aq} \\langle x-y\\rangle^{-(d+1)}\\, dy \n\t\\le C  \\int_{|y|< |x-y|} |y|^{-aq} \\langle y\\rangle^{-(d+1)} \\, dy \n\t< \\infty,\n\\end{align*}\nif $aq<d.$ Moreover, we have \n\\begin{equation*}\n\tI_2(x) \\le C \\int_{|y|> |x-y|} |x-y|^{-(a-b)q}  g(x-y)^q \\, dy \n\t\\le C \\int_{\\mathbb{R}^d} |y|^{-(a-b)q}  g(y)^q \\, dy < \\infty,\n\\end{equation*}\nif $a\\ge 0$ and $(a-b)q <d.$ \nThus, $I(x) < \\infty$ uniformly with respect to $x\\in\\mathbb{R}^d.$ \nThe proof for the second inequality is similar so we omit it. \n\\end{proof}\n\nWe recall the following elementary characterization of $L^1(\\mathbb{R}^d)$-functions. \n\\begin{prop}\t\\label{p:L1sg.dcy}\nIf $f \\in L^1(\\mathbb{R}^d),$ then \n\\begin{equation}\\nonumber\n\t\\liminf_{|x|\\to 0} |x|^d |f(x)| = \\liminf_{|x|\\to \\infty} |x|^d |f(x)| =0.\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nWe show the contrapositive. Suppose that \n$\\displaystyle\\liminf_{|x|\\to0} |x|^d |f(x)| = c>0.$ \nThen there exists some positive $\\delta$ such that $\\frac{c}2 \\le |x|^d |f(x)|$ \nfor $|x|\\le \\delta.$ Thus, \n\\begin{equation}\\nonumber\n\t\\int_{|x|\\le\\delta} |f(x)| dx \\ge c \\int_0^\\delta r^{-1} \\,dr\n\t= c \\left[ \\log r\\right]_{0}^r = + \\infty,\n\\end{equation}\nwhich implies $f\\notin L^1(\\mathbb{R}^d).$ The second equality is similarly proved. \n\\end{proof}\nAs a corollary, we have the following. \n\\begin{cor}\\label{c:wLp.sg.dcy}\nLet $s\\in\\mathbb{R}$ and $p \\in [1,\\infty].$ \nIf $f \\in L^{p}_{s}(\\mathbb{R}^d),$ then \n\\begin{equation}\\nonumber\n\t\\liminf_{|x|\\to 0} |x|^{s+\\frac{d}{p}} |g(x)| \n\t= \\liminf_{|x|\\to \\infty} |x|^{s+\\frac{d}{p}} |g(x)| =0.\n\\end{equation}\n\\end{cor}\n\nFinally, we give a proof of the fact that the $L^q_{\\tilde{s}}(\\mathbb{R}^d)$-mild solutions \nalso satisfy the equation \\eqref{HH} in the distributional sense. \n\\begin{lem}\t\\label{mildweak}\nWe assume the same assumptions as in Theorem \\ref{t:HH.LWP} (resp. Theorem \\ref{t:HH.LWP.sub}). Let $u$ be a $L^q_{\\tilde{s}}(\\mathbb{R}^d)$-mild solution on $[0,T)$ in the sense of Definition \\ref{def:sol-A}. \nThen $u$ is a weak solution in the sense of Definition \\ref{d:w.sol}.\n\\end{lem}\n\\begin{proof\nWe prove the critical case only, since the subcritical case can be treated in the similar manner. Let $T>0$ and $u$ be an $L^q_{s_c}(\\mathbb{R}^d)$-mild solution on $[0,T]$. First we prove $u\\in L^{\\alpha}(0,T;L^{\\alpha}_{\\frac{\\gamma}{\\alpha},loc}(\\mathbb{R}^d))$.\nLet $\\Omega\\subset \\mathbb{R}^d$ be a compact subset of $\\mathbb{R}^d$. We also assume that $q>\\alpha$ since the case $q=\\alpha$ can be treated in the similar manner with a slight modification. Since $s_c-2\\le s<(d+\\gamma)\/\\alpha-d\/q$, by the H\\\"older inequality, the following estimates hold: \n\\begin{align*}\n   \\|u\\|_{L^{\\alpha}(0,T;L^{\\alpha}_{\\frac{\\gamma}{\\alpha}}(\\Omega))}^{\\alpha}\n   &\\le \\int_0^T\\left(\\int_{\\Omega}|x|^{\\frac{q(\\gamma-\\alpha s)}{q-\\alpha}}dx\\right)^{\\frac{q}{q-\\alpha}}\\|u(t)\\|_{L_s^q} \\, dt \\\\\n   &\\le C\\int_0^Tt^{\\frac{s-s_c}{2}}\\,dt \\, \\|u\\|_{\\mathcal{K}^s(T)}<\\infty,\n\\end{align*}\nwhich implies that $u$ belongs to $L_t^{\\alpha}(0,T;L^{\\alpha}_{\\frac{\\gamma}{\\alpha},loc}(\\mathbb{R}^d))$. Next we prove that $u$ satisfies the weak form (\\ref{weak}). Let $\\eta\\in C^{1,2}([0,T]\\times\\mathbb{R}^d)$ be such that for any $t\\in [0,T]$, $\\operatorname{supp} \\eta(t, \\cdot)$ is compact. Let $T'\\in (0,T)$. Since $C_0^{\\infty}(\\mathbb{R}^d)$ is dense in $L^q_{s_c}(\\mathbb{R}^d)$ thanks to Proposition \\ref{p:wL.sp}, there exists a sequence $\\{u_{0j}\\}\\subset C_0^{\\infty}(\\mathbb{R}^d)$ such that the following identity holds:\n\\[\n    \\lim_{j\\rightarrow\\infty}\\|u_0-u_{0j}\\|_{L^q_{s_c}}=0.\n\\]\nBy this identity and the integration by parts, we can prove the following identity:\n\\begin{align*}\n   \\int_{[0,T']\\times\\mathbb{R}^d}&(e^{t\\Delta}u_0)(x)(\\Delta\\eta+\\partial_t\\eta)(t,x)\\,dxdt \\\\\n   &=\\int_{\\mathbb{R}^d}(e^{T'\\Delta}u_0)(x)\\, \\eta(T',x)\\,dx-\\int_{\\mathbb{R}^d}u_0(x)\\eta(0,x)\\,dx.\n\\end{align*}\nThus it suffices to prove the identity\n\\begin{equation}\t\\label{weak1}\n   \\int_{[0,T']\\times\\mathbb{R}^d}N(u(t,x))(\\Delta\\eta+\\partial_t\\eta)(t,x) \\,dxdt\n   =-\\int_{[0,T']\\times\\mathbb{R}^d}|x|^{\\gamma}F(u(t,x))\\eta(t,x) \\,dxdt,\n\\end{equation}\nwhere $N$ is defined by (\\ref{mapN}). We write $G(t,x):=|x|^{\\gamma}F(u(t,x))$. Then we can express $N(u)$ as\n\\[\nN(u)=\\int_0^te^{(t-\\tau)\\Delta}G(\\tau)\\,d\\tau.\n\\]\nMoreover, the equality\n\\[\n    \\sup_{t\\in [0,T]}t^{\\frac{(s_c-s)\\alpha}{2}}\\|G(t)\\|_{L_{\\sigma}^{\\frac{q}{\\alpha}}}=\\|u\\|_{\\mathcal{K}^s(T)}^{\\alpha}<\\infty\n\\]\nis valid, where $\\sigma:=\\alpha s-\\gamma$. Since the time interval $[0,T]$ is compact, by using mollifiers with respect to the time variable and the space variables, we can find $\\{G_j\\}\\subset C_0^{\\infty}([0,\\infty)\\times \\mathbb{R}^d)$ such that \n\\begin{equation}\t\\label{appro1}\n    \\lim_{j\\rightarrow\\infty}\\sup_{t\\in [0,T]}t^{\\frac{(s_c-s)\\alpha}{2}}\\|G(t)-G_j(t)\\|_{L^{\\frac{q}{\\alpha}}_{\\sigma}}=0.\n\\end{equation}\nWe define a sequence $\\{N_j\\}$ as\n\\[\n    N_j(t,x):=\\int_0^{t}e^{(t-\\tau)\\Delta}G_j(\\tau,x) \\, d\\tau.\n\\]\nIn a similar manner as the proof of Theorem \\ref{t:HH.LWP}, we can prove that\n\\[\n    \\|N_j-N(u)\\|_{\\mathcal{K}^s(T)}\\le C\\sup_{t\\in [0,T]}t^{\\frac{(s_c-s)\\alpha}{2}}\\|G_j(t)-G(t)\\|_{L_{\\sigma}^{\\frac{q}{\\alpha}}}\\rightarrow 0\n\\]\nas $j\\rightarrow \\infty$. By this fact, we deduce that \n\\[\n   \\text{R.H.S of (\\ref{weak1})}\n   =\\lim_{j\\rightarrow\\infty}\\int_{[0,T']\\times\\mathbb{R}^d} N_j(t,x)(\\Delta\\eta+\\partial_t\\eta)(t,x) \\,dxdt.\n\\]\nSince $G_j$ is smooth, so is $N_j$ and hence, by the integration by parts, the identity\n\\[\n   \\int_{[0,T']\\times\\mathbb{R}^d}N_j(t,x)(\\Delta\\eta+\\partial_t\\eta)(t,x)\\,dx\\,dt\n   \t=\\int_{[0,T']\\times\\mathbb{R}^d}G_j(t,x)\\eta(t,x) \\,dx\\,dt.\n\\]\nholds for any $j$. By taking the limit $j\\rightarrow\\infty$ in the right-hand side and (\\ref{appro1}), we have\n\\[\n   \\lim_{j\\rightarrow\\infty}\\int_{[0,T']\\times\\mathbb{R}^d}\n   N_j(t,x)(\\Delta\\eta+\\partial_t\\eta)(t,x) \\,dxdt\n   =\\int_{[0,T']\\times\\mathbb{R}^d}G(t,x)\\eta(t,x)\\,dxdt.\n\\]\nThus we obtain (\\ref{weak1}), which completes the proof of the lemma.\n\\end{proof}\n\n\\section*{Acknowledgement}\n\\par\nThe first author is supported by Grant-in-Aid for Young Scientists (B) \n(No. 17K14216) and Challenging Research (Pioneering) (No.17H06199), \nJapan Society for the Promotion of Science. \nThe second author is supported by JST CREST (No. JPMJCR1913), Japan and \nthe Grant-in-Aid for Scientific Research (B) (No.18H01132) and \nYoung Scientists Research (No.19K14581), JSPS.\nThe third author is supported by Grant-in-Aid for JSPS Fellows \n(No.19J00206), JSPS.\n\n\\begin{bibdiv}\n \\begin{biblist}[\\normalsize]\n \n\n\\bib{BenTayWei2017}{article}{\n   author={Ben Slimene, B.},\n   author={Tayachi, S.},\n   author={Weissler, F. B.},\n   title={Well-posedness, global existence and large time behavior for\n   Hardy-H\\'enon parabolic equations},\n   journal={Nonlinear Anal.},\n   volume={152},\n   date={2017},\n   pages={116--148},\n}\n\n\\bib{Ben2019}{article}{\n   author={Ben Slimene, B.},\n   title={Asymptotically self-similar global solutions for Hardy-H\\'{e}non\n   parabolic systems},\n   journal={Differ. Equ. Appl.},\n   volume={11},\n   date={2019},\n   number={4},\n   pages={439--462},\n}\n\n\n\n\n\n\n\n\n\n\n\\bib{Chi2019}{article}{\n   author={Chikami, N.},\n   title={Composition estimates and well-posedness for Hardy-H\\'{e}non parabolic\n   equations in Besov spaces},\n   journal={J. Elliptic Parabol. Equ.},\n   volume={5},\n   date={2019},\n   number={2},\n   pages={215--250},\n}\n\n\\bib{CIT-arxiv}{article}{\n   author={Chikami, N.},\n   author={Ikeda, M.},\n   author={Taniguchi, K.},\n   title={Well-posedness and global dynamics for the critical Hardy-Sobolev parabolic equation},\n   journal={arXiv:2009.07108v2},\n   date={2019},\n}\n\n\n\n\n\\bib{GW-2005}{article}{\n   author={Gazzola, F.},\n   author={Weth, T.},\n   title={Finite time blow-up and global solutions for semilinear parabolic\n   equations with initial data at high energy level},\n   journal={Differential Integral Equations},\n   volume={18},\n   date={2005},\n   number={9},\n   pages={961--990},\n}\n\n\\bib{GhoMor2013}{book}{\n   author={Ghoussoub, N.},\n   author={Moradifam, A.},\n   title={Functional inequalities: new perspectives and new applications},\n   series={Mathematical Surveys and Monographs},\n   volume={187},\n   publisher={American Mathematical Society, Providence, RI},\n   date={2013},\n   pages={xxiv+299},\n}\n\n\n\\bib{Gig86}{article}{\n   author={Giga, M.},\n   title={Solutions for semilinear parabolic equations in $L^p$ and\n   regularity of weak solutions of the Navier-Stokes system},\n   journal={J. Differential Equations},\n   volume={62},\n   date={1986},\n   number={2},\n   pages={186--212},\n}\n\n\n\n\n\\bib{H-1973}{article}{\n   author={H\\'enon, M.},\n   title={Numerical experiments on the stability of spherical stellar systems},\n   journal={Astron. Astrophys},\n   volume={24},\n   date={1973},\n   pages={229--238},\n}\n\n\\bib{Hir2008}{article}{\n   author={Hirose, M.},\n   title={Existence of global solutions for a semilinear parabolic Cauchy\n   problem},\n   journal={Differential Integral Equations},\n   volume={21},\n   date={2008},\n   number={7-8},\n   pages={623--652},\n}\n\n\\bib{HisIsh2018}{article}{\n   author={Hisa, K.},\n   author={Ishige, K.},\n   title={Existence of solutions for a fractional semilinear parabolic\n   equation with singular initial data},\n   journal={Nonlinear Anal.},\n   volume={175},\n   date={2018},\n   pages={108--132},\n}\n\n\\bib{HisTak-arxiv}{article}{\n   author={Hisa, K.},\n   author={Takahashi, J.},\n   title={Optimal singularities of initial data for solvability of the Hardy parabolic equation},\n   journal={arXiv:2102.04618},\n   date={2021},\n}\n\n\\bib{II-15}{article}{\n   author={Ikeda, M.},\n   author={Inui, T.},\n   title={Some non-existence results for the semilinear Schr\\\"odinger equation without gauge invariance},\n   journal={J. Math. Anal. Appl.},\n   volume={425},\n   date={2015},\n   pages={758--773},\n}\n\n\\bib{IT-arxiv}{article}{\n   author={Ikeda, M.},\n   author={Taniguchi, K.},\n   title={Global well-posedness, dissipation and blow up for semilinear heat equations in energy spaces associated with self-adjoint operators},\n   journal={arXiv:1902.01016v3},\n   date={2019},\n}\n\n\n\n\n\\bib{Ish2008}{article}{\n   author={Ishiwata, M.},\n   title={Asymptotic behavior of strong solutions for nonlinear parabolic\n   equations with critical Sobolev exponent},\n   journal={Adv. Differential Equations},\n   volume={13},\n   date={2008},\n   number={3-4},\n   pages={349--366},\n}\n\n\n\n\n\n\n\n\\bib{Maj-arxiv}{article}{\n   author={Majdoub, M.},\n   title={Well-posedness and blow-up for an inhomogeneous semilinear parabolic equation},\n   journal={arXiv:2008.01290v3},\n   date={2021},\n}\n\n\n\\bib{NakTomYab2004}{article}{\n   author={Nakai, E.},\n   author={Tomita, N.},\n   author={Yabuta, K.},\n   title={Density of the set of all infinitely differentiable functions with\n   compact support in weighted Sobolev spaces},\n   journal={Sci. Math. Jpn.},\n   volume={60},\n   date={2004},\n   number={1},\n}\n\n\n\\bib{OkaTsu2016}{article}{\n   author={Okabe, T.},\n   author={Tsutsui, Y.},\n   title={Navier-Stokes flow in the weighted Hardy space with applications\n   to time decay problem},\n   journal={J. Differential Equations},\n   volume={261},\n   date={2016},\n   number={3},\n   pages={1712--1755},\n}\n\n\\bib{Ouh2005}{book}{\n   author={Ouhabaz, E. M.},\n   title={Analysis of heat equations on domains},\n   series={London Mathematical Society Monographs Series},\n   volume={31},\n   publisher={Princeton University Press, Princeton, NJ},\n   date={2005},\n}\n\n\\bib{Pin1997}{article}{\n   author={Pinsky, R.~G.~},\n   title={Existence and nonexistence of global solutions for $u_t=\\Delta\n   u+a(x)u^p$ in ${\\bf R}^d$},\n   journal={J. Differential Equations},\n   volume={133},\n   date={1997},\n   number={1},\n   pages={152--177},\n   issn={0022-0396},\n}\n\n\n\n\\bib{Qi1998}{article}{\n   author={Qi, Y.},\n   title={The critical exponents of parabolic equations and blow-up in ${\\bf\n   R}^n$},\n   journal={Proc. Roy. Soc. Edinburgh Sect. A},\n   volume={128},\n   date={1998},\n   number={1},\n   pages={123--136},\n}\n\n\n\n\n\n\\bib{SnoTayWei2001}{article}{\n   author={Snoussi, S.},\n   author={Tayachi, S.},\n   author={Weissler, F. B.},\n   title={Asymptotically self-similar global solutions of a general\n   semilinear heat equation},\n   journal={Math. Ann.},\n   volume={321},\n   date={2001},\n   number={1},\n   pages={131--155},\n}\n\n\n\n\n\\bib{Tay2020}{article}{\n   author={Tayachi, S.},\n   title={Uniqueness and non-uniqueness of solutions for critical\n   Hardy-H\\'{e}non parabolic equations},\n   journal={J. Math. Anal. Appl.},\n   volume={488},\n   date={2020},\n   number={1},\n}\n\n\\bib{Tsu2011}{article}{\n   author={Tsutsui, Y.},\n   title={The Navier-Stokes equations and weak Herz spaces},\n   journal={Adv. Differential Equations},\n   volume={16},\n   date={2011},\n   number={11-12},\n   pages={1049--1085},\n}\n\n\\bib{Tsu2014}{article}{\n   author={Tsutsui, Y.},\n   title={An application of weighted Hardy spaces to the Navier-Stokes\n   equations},\n   journal={J. Funct. Anal.},\n   volume={266},\n   date={2014},\n   number={3},\n   pages={1395--1420},\n}\n\n\n\n\\bib{Wan1993}{article}{\n   author={Wang, X.},\n   title={On the Cauchy problem for reaction-diffusion equations},\n   journal={Trans. Amer. Math. Soc.},\n   volume={337},\n   date={1993},\n   number={2},\n   pages={549--590},\n}\n\n\\bib{Wei1979}{article}{\n   author={Weissler, F. B.},\n   title={Semilinear evolution equations in Banach spaces},\n   journal={J. Functional Analysis},\n   volume={32},\n   date={1979},\n   number={3},\n   pages={277--296},\n}\n\n\\bib{Wei1980}{article}{\n   author={Weissler, F. B.},\n   title={Local existence and nonexistence for semilinear parabolic\n   equations in $L^{p}$},\n   journal={Indiana Univ. Math. J.},\n   volume={29},\n   date={1980},\n   number={1},\n   pages={79--102},\n}\n\n \\end{biblist}\n\\end{bibdiv} \n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}